Python supports a wide range of arithmetic operators that you can use when working with numbers in your code. One of these operators is the **modulo operator** (`%`

), which returns the remainder of dividing two numbers.

**In this tutorial, you’ll learn:**

- How
**modulo**works in mathematics - How to use the Python modulo operator with different
**numeric types** - How Python calculates the results of a
**modulo operation** - How to override
in your classes to use them with the modulo operator`.__mod__()`

- How to use the Python modulo operator to solve
**real-world problems**

The Python modulo operator can sometimes be overlooked. But having a good understanding of this operator will give you an invaluable tool in your Python tool belt.

**Free Bonus:** Click here to get a Python Cheat Sheet and learn the basics of Python 3, like working with data types, dictionaries, lists, and Python functions.

## Modulo in Mathematics#

The term **modulo** comes from a branch of mathematics called modular arithmetic. Modular arithmetic deals with integer arithmetic on a circular number line that has a fixed set of numbers. All arithmetic operations performed on this number line will wrap around when they reach a certain number called the **modulus**.

A classic example of modulo in modular arithmetic is the twelve-hour clock. A twelve-hour clock has a fixed set of values, from 1 to 12. When counting on a twelve-hour clock, you count up to the modulus 12 and then wrap back to 1. A twelve-hour clock can be classified as “modulo 12,” sometimes shortened to “mod 12.”

The modulo operator is used when you want to compare a number with the modulus and get the equivalent number constrained to the range of the modulus.

For example, say you want to determine what time it would be nine hours after 8:00 a.m. On a twelve-hour clock, you can’t simply add 9 to 8 because you would get 17. You need to take the result, 17, and use `mod`

to get its equivalent value in a twelve-hour context:

```
8 o'clock + 9 = 17 o'clock
17 mod 12 = 5
```

`17 mod 12`

returns `5`

. This means that nine hours past 8:00 a.m. is 5:00 p.m. You determined this by taking the number `17`

and applying it to a `mod 12`

context.

Now, if you think about it, `17`

and `5`

are equivalent in a `mod 12`

context. If you were to look at the hour hand at 5:00 and 17:00, it would be in the same position. Modular arithmetic has an equation to describe this relationship:

```
a ≡ b (mod n)
```

This equation reads “`a`

and `b`

are congruent modulo `n`

.” This means that `a`

and `b`

are equivalent in `mod n`

as they have the same remainder when divided by `n`

. In the above equation, `n`

is the **modulus** for both `a`

and `b`

. Using the values `17`

and `5`

from before, the equation would look like this:

```
17 ≡ 5 (mod 12)
```

This reads “`17`

and `5`

are congruent modulo `12`

.” `17`

and `5`

have the same remainder, `5`

, when divided by `12`

. So in `mod 12`

, the numbers `17`

and `5`

are equivalent.

You can confirm this using division:

```
17 / 12 = 1 R 5
5 / 12 = 0 R 5
```

Both of the operations have the same remainder, `5`

, so they’re equivalent modulo `12`

.

Now, this may seem like a lot of math for a Python operator, but having this knowledge will prepare you to use the modulo operator in the examples later in this tutorial. In the next section, you’ll look at the basics of using the Python modulo operator with the numeric types `int`

and `float`

.

## Python Modulo Operator Basics#

The modulo operator, like the other arithmetic operators, can be used with the numeric types `int`

and `float`

. As you’ll see later on, it can also be used with other types like `math.fmod()`

, `decimal.Decimal`

, and your own classes.

### Modulo Operator With `int`

#

Most of the time you’ll use the modulo operator with integers. The modulo operator, when used with two positive integers, will return the remainder of standard Euclidean division:

```
>>> 15 % 4
3
>>> 17 % 12
5
>>> 240 % 13
6
>>> 10 % 16
10
```

Be careful! Just like with the division operator (`/`

), Python will return a `ZeroDivisionError`

if you try to use the modulo operator with a divisor of `0`

:

```
>>> 22 % 0
ZeroDivisionError: integer division or modulo by zero
```

Next, you’ll take a look at using the modulo operator with a `float`

.

### Modulo Operator With `float`

#

Similar to `int`

, the modulo operator used with a `float`

will return the remainder of division, but as a `float`

value:

```
>>> 12.5 % 5.5
1.5
>>> 17.0 % 12.0
5.0
```

An alternative to using a `float`

with the modulo operator is to use `math.fmod()`

to perform modulo operations on `float`

values:

```
>>> import math
>>> math.fmod(12.5, 5.5)
1.5
>>> math.fmod(8.5, 2.5)
1.0
```

The official Python docs suggest using `math.fmod()`

over the Python modulo operator when working with `float`

values because of the way `math.fmod()`

calculates the result of the modulo operation. If you’re using a negative operand, then you may see different results between `math.fmod(x, y)`

and `x % y`

. You’ll explore using the modulo operator with negative operands in more detail in the next section.

Just like other arithmetic operators, the modulo operator and `math.fmod()`

may encounter rounding and precision issues when dealing with floating-point arithmetic:

```
>>> 13.3 % 1.1
0.09999999999999964
>>> import math
>>> math.fmod(13.3, 1.1)
0.09999999999999964
```

If maintaining floating-point precision is important to your application, then you can use the modulo operator with `decimal.Decimal`

. You’ll look at this later in this tutorial.

### Modulo Operator With a Negative Operand#

All modulo operations you’ve seen up to this point have used two positive operands and returned predictable results. When a negative operand is introduced, things get more complicated.

As it turns out, the way that computers determine the result of a modulo operation with a negative operand leaves ambiguity as to whether the remainder should take the sign of the **dividend** (the number being divided) or the sign of the **divisor** (the number by which the dividend is divided). Different programming languages handle this differently.

For example, in JavaScript, the remainder will take the sign of the dividend:

```
8 % -3 = 2
```

The remainder in this example, `2`

, is positive since it takes the sign of the dividend, `8`

. In Python and other languages, the remainder will take the sign of the divisor instead:

```
8 % -3 = -1
```

Here you can see that the remainder, `-1`

, takes the sign of the divisor, `-3`

.

You may be wondering why the remainder in JavaScript is `2`

and the remainder in Python is `-1`

. This has to do with how different languages determine the outcome of a modulo operation. Languages in which the remainder takes the sign of the dividend use the following equation to determine the remainder:

```
r = a - (n * trunc(a/n))
```

There are three variables this equation:

is the remainder.`r`

is the dividend.`a`

is the divisor.`n`

`trunc()`

in this equation means that it uses **truncated division**, which will always round a negative number toward zero. For more clarification, see the steps of the modulo operation below using `8`

as the dividend and `-3`

as the divisor:

```
r = 8 - (-3 * trunc(8/-3))
r = 8 - (-3 * trunc(-2.666666666667))
r = 8 - (-3 * -2) # Rounded toward 0
r = 8 - 6
r = 2
```

Here you can see how a language like JavaScript gets the remainder `2`

. Python and other languages in which the remainder takes the sign of the divisor use the following equation:

```
r = a - (n * floor(a/n))
```

`floor()`

in this equation means that it uses **floor division**. With positive numbers, floor division will return the same result as truncated division. But with a negative number, floor division will round the result down, away from zero:

```
r = 8 - (-3 * floor(8/-3))
r = 8 - (-3 * floor(-2.666666666667))
r = 8 - (-3 * -3) # Rounded away from 0
r = 8 - 9
r = -1
```

Here you can see that the result is `-1`

.

Now that you understand where the difference in the remainder comes from, you may be wondering why this matters if you only use Python. Well, as it turns out, not all modulo operations in Python are the same. While the modulo used with the `int`

and `float`

types will take the sign of the divisor, other types will not.

You can see an example of this when you compare the results of `8.0 % -3.0`

and `math.fmod(8.0, -3.0)`

:

```
>>> 8.0 % -3
-1.0
>>> import math
>>> math.fmod(8.0, -3.0)
2.0
```

`math.fmod()`

takes the sign of the dividend using truncated division, whereas `float`

uses the sign of the divisor. Later in this tutorial, you’ll see another Python type that uses the sign of the dividend, `decimal.Decimal`

.

### Modulo Operator and `divmod()`

#

Python has the built-in function `divmod()`

, which internally uses the modulo operator. `divmod()`

takes two parameters and returns a tuple containing the results of floor division and modulo using the supplied parameters.

Below is an example of using `divmod()`

with `37`

and `5`

:

```
>>> divmod(37, 5)
(7, 2)
>>> 37 // 5
7
>>> 37 % 5
2
```

You can see that `divmod(37, 5)`

returns the tuple `(7, 2)`

. The `7`

is the result of the floor division of `37`

and `5`

. The `2`

is the result of `37`

modulo `5`

.

Below is an example in which the second parameter is a negative number. As discussed in the previous section, when the modulo operator is used with an `int`

, the remainder will take the sign of the divisor:

```
>>> divmod(37, -5)
(-8, -3)
>>> 37 // -5
-8
>>> 37 % -5
-3 # Result has the sign of the divisor
```

Now that you’ve had a chance to see the modulo operator used in several scenarios, it’s important to take a look at how Python determines the precedence of the modulo operator when used with other arithmetic operators.

### Modulo Operator Precedence#

Like other Python operators, there are specific rules for the modulo operator that determine its precedence when evaluating expressions. The modulo operator (`%`

) shares the same level of precedence as the multiplication (`*`

), division (`/`

), and floor division (`//`

) operators.

Take a look at an example of the modulo operator’s precedence below:

```
>>> 4 * 10 % 12 - 9
-5
```

Both the multiplication and modulo operators have the same level of precedence, so Python will evaluate them from left to right. Here are the steps for the above operation:

is evaluated, resulting in`4 * 10`

`40 % 12 - 9`

.is evaluated, resulting in`40 % 12`

`4 - 9`

.is evaluated, resulting in`4 - 9`

`-5`

.

If you want to override the precedence of other operators, then you can use parentheses to surround the operation you want to be evaluated first:

```
>>> 4 * 10 % (12 - 9)
1
```

In this example, `(12 - 9)`

is evaluated first, followed by `4 * 10`

and finally `40 % 3`

, which equals `1`

.

## Python Modulo Operator in Practice#

Now that you’ve gone through the basics of the Python modulo operator, you’ll look at some examples of using it to solve real-world programming problems. At times, it can be hard to determine when to use the modulo operator in your code. The examples below will give you an idea of the many ways it can be used.

### How to Check if a Number Is Even or Odd#

In this section, you’ll see how you can use the modulo operator to determine if a number is even or odd. Using the modulo operator with a modulus of `2`

, you can check any number to see if it’s evenly divisible by `2`

. If it is evenly divisible, then it’s an even number.

Take a look at `is_even()`

which checks to see if the `num`

parameter is even:

```
def is_even(num):
return num % 2 == 0
```

Here `num % 2`

will equal `0`

if `num`

is even and `1`

if `num`

is odd. Checking against `0`

will return a Boolean of `True`

or `False`

based on whether or not `num`

is even.

Checking for odd numbers is quite similar. To check for an odd number, you invert the equality check:

```
def is_odd(num):
return num % 2 != 0
```

This function will return `True`

if `num % 2`

does not equal `0`

, meaning that there’s a remainder proving `num`

is an odd number. Now, you may be wondering if you could use the following function to determine if `num`

is an odd number:

```
def is_odd(num):
return num % 2 == 1
```

The answer to this question is yes *and* no. Technically, this function will work with the way Python calculates modulo with integers. That said, you should avoid comparing the result of a modulo operation with `1`

as not all modulo operations in Python will return the same remainder.

You can see why in the following examples:

```
>>> -3 % 2
1
>>> 3 % -2
-1
```

In the second example, the remainder takes the sign of the negative divisor and returns `-1`

. In this case, the Boolean check `3 % -2 == 1`

would return `False`

.

However, if you compare the modulo operation with `0`

, then it doesn’t matter which operand is negative. The result will always be `True`

when it’s an even number:

```
>>> -2 % 2
0
>>> 2 % -2
0
```

If you stick to comparing a Python modulo operation with `0`

, then you shouldn’t have any problems checking for even and odd numbers or any other multiples of a number in your code.

In the next section, you’ll take a look at how you can use the modulo operator with loops to control the flow of your program.

### How to Run Code at Specific Intervals in a Loop#

With the Python modulo operator, you can run code at specific intervals inside a loop. This is done by performing a modulo operation with the current index of the loop and a modulus. The modulus number determines how often the interval-specific code will run in the loop.

Here’s an example:

```
def split_names_into_rows(name_list, modulus=3):
for index, name in enumerate(name_list, start=1):
print(f"{name:-^15} ", end="")
if index % modulus == 0:
print()
print()
```

This code defines `split_names_into_rows()`

, which takes two parameters. `name_list`

is a list of names that should be split into rows. `modulus`

sets a modulus for the operation, effectively determining how many names should be in each row. `split_names_into_rows()`

will loop over `name_list`

and start a new row after it hits the `modulus`

value.

Before breaking down the function in more detail, take a look at it in action:

```
>>> names = ["Picard", "Riker", "Troi", "Crusher", "Worf", "Data", "La Forge"]
>>> split_names_into_rows(names)
----Picard----- -----Riker----- -----Troi------
----Crusher---- -----Worf------ -----Data------
---La Forge----
```

As you can see, the list of names has been split into three rows, with a maximum of three names in each row. `modulus`

defaults to `3`

, but you can specify any number:

```
>>> split_names_into_rows(names, modulus=4)
----Picard----- -----Riker----- -----Troi------ ----Crusher----
-----Worf------ -----Data------ ---La Forge----
>>> split_names_into_rows(names, modulus=2)
----Picard----- -----Riker-----
-----Troi------ ----Crusher----
-----Worf------ -----Data------
---La Forge----
>>> split_names_into_rows(names, modulus=1)
----Picard-----
-----Riker-----
-----Troi------
----Crusher----
-----Worf------
-----Data------
---La Forge----
```

Now that you’ve seen the code in action, you can break down what it’s doing. First, it uses `enumerate`

to iterate over `name_list`

, assigning the current item in the list to `name`

and a count value to `index`

. You can see that the optional `start`

argument for `enumerate`

is set to `1`

. This means that the `index`

count will start at `1`

instead of `0`

:

```
for index, name in enumerate(name_list, start=1):
```

Next, inside the loop, the function calls `print()`

to output `name`

to the current row. The `end`

parameter for `print()`

is an empty string (`""`

) so it won’t output a newline at the end of the string. An f-string is passed to `print()`

, which uses the string output formatting syntax that Python provides:

```
print(f"{name:-^15} ", end="")
```

Without getting into too much detail, the `:-^15`

syntax tells `print()`

to do the following:

- Output at least
`15`

characters, even if the string is shorter than 15 characters. - Center align the string.
- Fill any space on the right or left of the string with the hyphen character (
`-`

).

Now that the name has been printed to the row, take a look at the main part of `split_names_into_rows()`

:

```
if index % modulus == 0:
print()
```

This code takes the current iteration `index`

and, using the modulo operator, compares it with `modulus`

. If the result equals `0`

, then it can run interval-specific code. In this case, the function calls `print()`

to add a newline, which starts a new row.

The above code is only one example. Using the pattern `index % modulus == 0`

allows you to run different code at specific intervals in your loops. In the next section, you’ll take this concept a bit further and look at cyclic iteration.

### How to Create Cyclic Iteration#

**Cyclic iteration** describes a type of iteration that will reset once it gets to a certain point. Generally, this type of iteration is used to restrict the index of the iteration to a certain range.

You can use the modulo operator to create cyclic iteration. Take a look at an example using the `turtle`

library to draw a shape:

```
import turtle
import random
def draw_with_cyclic_iteration():
colors = ["green", "cyan", "orange", "purple", "red", "yellow", "white"]
turtle.bgcolor("gray8") # Hex: #333333
turtle.pendown()
turtle.pencolor(random.choice(colors)) # First color is random
i = 0 # Initial index
while True:
i = (i + 1) % 6 # Update the index
turtle.pensize(i) # Set pensize to i
turtle.forward(225)
turtle.right(170)
# Pick a random color
if i == 0:
turtle.pencolor(random.choice(colors))
```

The above code uses an infinite loop to draw a repeating star shape. After every six iterations, it changes the color of the pen. The pen size increases with each iteration until `i`

is reset back to `0`

. If you run the code, then you should get something similar to this:

The important parts of this code are highlighted below:

```
import turtle
import random
def draw_with_cyclic_iteration():
colors = ["green", "cyan", "orange", "purple", "red", "yellow", "white"]
turtle.bgcolor("gray8") # Hex: #333333
turtle.pendown()
turtle.pencolor(random.choice(colors))
i = 0 # Initial index
while True:
i = (i + 1) % 6 # Update the index
turtle.pensize(i) # Set pensize to i
turtle.forward(225)
turtle.right(170)
# Pick a random color
if i == 0:
turtle.pencolor(random.choice(colors))
```

Each time through the loop, `i`

is updated based on the results of `(i + 1) % 6`

. This new `i`

value is used to increase the `.pensize`

with each iteration. Once `i`

reaches `5`

, `(i + 1) % 6`

will equal `0`

, and `i`

will reset back to `0`

.

You can see the steps of the iteration below for more clarification:

```
i = 0 : (0 + 1) % 6 = 1
i = 1 : (1 + 1) % 6 = 2
i = 2 : (2 + 1) % 6 = 3
i = 3 : (3 + 1) % 6 = 4
i = 4 : (4 + 1) % 6 = 5
i = 5 : (5 + 1) % 6 = 0 # Reset
```

When `i`

is reset back to `0`

, the `.pencolor`

changes to a new random color as seen below:

```
if i == 0:
turtle.pencolor(random.choice(colors))
```

The code in this section uses `6`

as the modulus, but you could set it to any number to adjust how many times the loop will iterate before resetting the value `i`

.

### How to Convert Units#

In this section, you’ll look at how you can use the modulo operator to convert units. The following examples take smaller units and convert them into larger units without using decimals. The modulo operator is used to determine any remainder that may exist when the smaller unit isn’t evenly divisible by the larger unit.

In this first example, you’ll convert inches into feet. The modulo operator is used to get the remaining inches that don’t evenly divide into feet. The floor division operator (`//`

) is used to get the total feet rounded down:

```
def convert_inches_to_feet(total_inches):
inches = total_inches % 12
feet = total_inches // 12
print(f"{total_inches} inches = {feet} feet and {inches} inches")
```

Here’s an example of the function in use:

```
>>> convert_inches_to_feet(450)
450 inches = 37 feet and 6 inches
```

As you can see from the output, `450 % 12`

returns `6`

, which is the remaining inches that weren’t evenly divided into feet. The result of `450 // 12`

is `37`

, which is the total number of feet by which the inches were evenly divided.

You can take this a bit further in this next example. `convert_minutes_to_days()`

takes an integer, `total_mins`

, representing a number of minutes and outputs the period of time in days, hours, and minutes:

```
def convert_minutes_to_days(total_mins):
days = total_mins // 1440
extra_minutes = total_mins % 1440
hours = extra_minutes // 60
minutes = extra_minutes % 60
print(f"{total_mins} = {days} days, {hours} hours, and {minutes} minutes")
```

Breaking this down, you can see that the function does the following:

- Determines the total number of evenly divisible days with
`total_mins // 1440`

, where`1440`

is the number of minutes in a day - Calculates any
`extra_minutes`

left over with`total_mins % 1440`

- Uses the
`extra_minutes`

to get the evenly divisible`hours`

and any extra`minutes`

You can see how it works below:

```
>>> convert_minutes_to_days(1503)
1503 = 1 days, 1 hours, and 3 minutes
>>> convert_minutes_to_days(3456)
3456 = 2 days, 9 hours, and 36 minutes
>>> convert_minutes_to_days(35000)
35000 = 24 days, 7 hours, and 20 minutes
```

While the above examples only deal with converting inches to feet and minutes to days, you could use any type of units with the modulo operator to convert a smaller unit into a larger unit.

**Note**: Both of the above examples could be modified to use `divmod()`

to make the code more succinct. If you remember, `divmod()`

returns a tuple containing the results of floor division and modulo using the supplied parameters.

Below, the floor division and modulo operators have been replaced with `divmod()`

:

```
def convert_inches_to_feet_updated(total_inches):
feet, inches = divmod(total_inches, 12)
print(f"{total_inches} inches = {feet} feet and {inches} inches")
```

As you can see, `divmod(total_inches, 12)`

returns a tuple, which is unpacked into `feet`

and `inches`

.

If you try this updated function, then you’ll receive the same results as before:

```
>>> convert_inches_to_feet(450)
450 inches = 37 feet and 6 inches
>>> convert_inches_to_feet_updated(450)
450 inches = 37 feet and 6 inches
```

You receive the same outcome, but now the code is more concise. You could update `convert_minutes_to_days()`

as well:

```
def convert_minutes_to_days_updated(total_mins):
days, extra_minutes = divmod(total_mins, 1440)
hours, minutes = divmod(extra_minutes, 60)
print(f"{total_mins} = {days} days, {hours} hours, and {minutes} minutes")
```

Using `divmod()`

, the function is easier to read than the previous version and returns the same result:

```
>>> convert_minutes_to_days(1503)
1503 = 1 days, 1 hours, and 3 minutes
>>> convert_minutes_to_days_updated(1503)
1503 = 1 days, 1 hours, and 3 minutes
```

Using `divmod()`

isn’t necessary for all situations, but it makes sense here as the unit conversion calculations use both floor division and modulo.

Now that you’ve seen how to use the modulo operator to convert units, in the next section you’ll look at how you can use the modulo operator to check for prime numbers.

### How to Determine if a Number Is a Prime Number#

In this next example, you’ll take a look at how you can use the Python modulo operator to check whether a number is a **prime number**. A prime number is any number that contains only two factors, `1`

and itself. Some examples of prime numbers are `2`

, `3`

, `5`

, `7`

, `23`

, `29`

, `59`

, `83`

, and `97`

.

The code below is an implementation for determining the primality of a number using the modulo operator:

```
def check_prime_number(num):
if num < 2:
print(f"{num} must be greater than or equal to 2 to be prime.")
return
factors = [(1, num)]
i = 2
while i * i <= num:
if num % i == 0:
factors.append((i, num//i))
i += 1
if len(factors) > 1:
print(f"{num} is not prime. It has the following factors: {factors}")
else:
print(f"{num} is a prime number")
```

This code defines `check_prime_number()`

, which takes the parameter `num`

and checks to see if it’s a prime number. If it is, then a message is displayed stating that `num`

is a prime number. If it’s not a prime number, then a message is displayed with all the factors of the number.

**Note:** The above code isn’t the most efficient way to check for prime numbers. If you’re interested in digging deeper, then check out the Sieve of Eratosthenes and Sieve of Atkin for examples of more performant algorithms for finding prime numbers.

Before you look more closely at the function, here are the results using some different numbers:

```
>>> check_prime_number(44)
44 is not prime. It has the following factors: [(1, 44), (2, 22), (4, 11)]
>>> check_prime_number(53)
53 is a prime number
>>> check_prime_number(115)
115 is not prime. It has the following factors: [(1, 115), (5, 23)]
>>> check_prime_number(997)
997 is a prime number
```

Digging into the code, you can see it starts by checking if `num`

is less than `2`

. Prime numbers can only be greater than or equal to `2`

. If `num`

is less than `2`

, then the function doesn’t need to continue. It will `print()`

a message and `return`

:

```
if num < 2:
print(f"{num} must be greater than or equal to 2 to be prime.")
return
```

If `num`

is greater than `2`

, then the function checks if `num`

is a prime number. To check this, the function iterates over all the numbers between `2`

and the square root of `num`

to see if any divide evenly into `num`

. If one of the numbers divides evenly, then a factor has been found, and `num`

can’t be a prime number.

Here’s the main part of the function:

```
factors = [(1, num)]
i = 2
while i * i <= num:
if num % i == 0:
factors.append((i, num//i))
i += 1
```

There’s a lot to unpack here, so let’s take it step by step.

First, a `factors`

list is created with the initial factors, `(1, num)`

. This list will be used to store any other factors that are found:

```
factors = [(1, num)]
```

Next, starting with `2`

, the code increments `i`

until it reaches the square root of `num`

. At each iteration, it compares `num`

with `i`

to see if it’s evenly divisible. The code only needs to check up to and including the square root of `num`

because it wouldn’t contain any factors above this:

```
i = 2
while i * i <= num:
if num % i == 0:
factors.append((i, num//i))
i += 1
```

Instead of trying to determine the square root of `num`

, the function uses a `while`

loop to see if `i * i <= num`

. As long as `i * i <= num`

, the loop hasn’t reached the square root of `num`

.

Inside the `while`

loop, the modulo operator checks if `num`

is evenly divisible by `i`

:

```
factors = [(1, num)]
i = 2 # Start the initial index at 2
while i * i <= num:
if num % i == 0:
factors.append((i, num//i))
i += 1
```

If `num`

is evenly divisible by `i`

, then `i`

is a factor of `num`

, and a tuple of the factors is added to the `factors`

list.

Once the `while`

loop is complete, the code checks to see if any additional factors were found:

```
if len(factors) > 1:
print(f"{num} is not prime. It has the following factors: {factors}")
else:
print(f"{num} is a prime number")
```

If more than one tuple exists in the `factors`

list, then `num`

can’t be a prime number. For nonprime numbers, the factors are printed out. For prime numbers, the function prints a message stating that `num`

is a prime number.

### How to Implement Ciphers#

The Python modulo operator can be used to create ciphers. A cipher is a type of algorithm for performing encryption and decryption on an input, usually text. In this section, you’ll look at two ciphers, the **Caesar cipher** and the **Vigenère cipher**.

#### Caesar Cipher#

The first cipher that you’ll look at is the Caesar cipher, named after Julius Caesar, who used it to secretly communicate messages. It’s a substitution cipher that uses letter substitution to encrypt a string of text.

The Caesar cipher works by taking a letter to be encrypted and shifting it a certain number of positions to the left or right in the alphabet. Whichever letter is in that position is used as the encrypted character. This same shift value is applied to all characters in the string.

For example, if the shift were `5`

, then `A`

would shift up five letters to become `F`

, `B`

would become `G`

, and so on. Below you can see the encryption process for the text `REALPYTHON`

with a shift of `5`

:

The resulting cipher is `WJFQUDYMTS`

.

Decrypting the cipher is done by reversing the shift. Both the encryption and decryption processes can be described with the following expressions, where `char_index`

is the index of the character in the alphabet:

```
encrypted_char_index = (char_index + shift) % 26
decrypted_char_index = (char_index - shift) % 26
```

This cipher uses the modulo operator to make sure that, when shifting a letter, the index will wrap around if the end of the alphabet is reached. Now that you know how this cipher works, take a look at an implementation:

```
import string
def caesar_cipher(text, shift, decrypt=False):
if not text.isascii() or not text.isalpha():
raise ValueError("Text must be ASCII and contain no numbers.")
lowercase = string.ascii_lowercase
uppercase = string.ascii_uppercase
result = ""
if decrypt:
shift = shift * -1
for char in text:
if char.islower():
index = lowercase.index(char)
result += lowercase[(index + shift) % 26]
else:
index = uppercase.index(char)
result += uppercase[(index + shift) % 26]
return result
```

This code defines a function called `caesar_cipher()`

, which has two required parameters and one optional parameter:

is the text to be encrypted or decrypted.`text`

is the number of positions to shift each letter.`shift`

is a Boolean to set if`decrypt`

`text`

should be decrypted.

`decrypt`

is included so that a single function can be used to handle both encryption and decryption. This implementation can handle only alphabetic characters, so the function first checks that `text`

is an alphabetic character in the ASCII encoding:

```
def caesar_cipher(text, shift, decrypt=False):
if not text.isascii() or not text.isalpha():
raise ValueError("Text must be ASCII and contain no numbers.")
```

The function then defines three variables to store the `lowercase`

ASCII characters, the `uppercase`

ASCII characters, and the results of the encryption or decryption:

```
lowercase = string.ascii_lowercase # "abcdefghijklmnopqrstuvwxyz"
uppercase = string.ascii_uppercase # "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
result = ""
```

Next, if the function is being used to decrypt `text`

, then it multiplies `shift`

by `-1`

to make it shift backward:

```
if decrypt:
shift = shift * -1
```

Finally, `caesar_cipher()`

loops over the individual characters in `text`

and performs the following actions for each `char`

:

- Check if
`char`

is lowercase or uppercase. - Get the
`index`

of the`char`

in either the`lowercase`

or`uppercase`

ASCII lists. - Add a
`shift`

to this`index`

to determine the index of the cipher character to use. - Use
`% 26`

to make sure the shift will wrap back to the start of the alphabet. - Append the cipher character to the
`result`

string.

After the loop finishes iterating over the `text`

value, the `result`

is returned:

```
for char in text:
if char.islower():
index = lowercase.index(char)
result += lowercase[(index + shift) % 26]
else:
index = uppercase.index(char)
result += uppercase[(index + shift) % 26]
return result
```

Here’s the full code again:

```
import string
def caesar_cipher(text, shift, decrypt=False):
if not text.isascii() or not text.isalpha():
raise ValueError("Text must be ASCII and contain no numbers.")
lowercase = string.ascii_lowercase
uppercase = string.ascii_uppercase
result = ""
if decrypt:
shift = shift * -1
for char in text:
if char.islower():
index = lowercase.index(char)
result += lowercase[(index + shift) % 26]
else:
index = uppercase.index(char)
result += uppercase[(index + shift) % 26]
return result
```

Now run the code in the Python REPL using the text `meetMeAtOurHideOutAtTwo`

with a shift of `10`

:

```
>>> caesar_cipher("meetMeAtOurHideOutAtTwo", 10)
woodWoKdYebRsnoYedKdDgy
```

The encrypted result is `woodWoKdYebRsnoYedKdDgy`

. Using this encrypted text, you can run the decryption to get the original text:

```
>>> caesar_cipher("woodWoKdYebRsnoYedKdDgy", 10, decrypt=True)
meetMeAtOurHideOutAtTwo
```

The Caesar cipher is fun to play around with for an introduction to cryptography. While the Caesar cipher is rarely used on its own, it’s the basis for more complex substitution ciphers. In the next section, you’ll look at one of the Caesar cipher’s descendants, the Vigenère cipher.

#### Vigenère Cipher#

The Vigenère cipher is a polyalphabetic substitution cipher. To perform its encryption, it employs a different Caesar cipher for each letter of the input text. The Vigenère cipher uses a keyword to determine which Caesar cipher should be used to find the cipher letter.

You can see an example of the encryption process in the following image. In this example, the input text `REALPYTHON`

is encrypted using the keyword `MODULO`

:

For each letter of the input text, `REALPYTHON`

, a letter from the keyword `MODULO`

is used to determine which Caesar cipher column should be selected. If the keyword is shorter than the input text, as is the case with `MODULO`

, then the letters of the keyword are repeated until all letters of the input text have been encrypted.

Below is an implementation of the Vigenère cipher. As you’ll see, the modulo operator is used twice in the function:

```
import string
def vigenere_cipher(text, key, decrypt=False):
if not text.isascii() or not text.isalpha() or not text.isupper():
raise ValueError("Text must be uppercase ASCII without numbers.")
uppercase = string.ascii_uppercase # "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
results = ""
for i, char in enumerate(text):
current_key = key[i % len(key)]
char_index = uppercase.index(char)
key_index = uppercase.index(current_key)
if decrypt:
index = char_index - key_index + 26
else:
index = char_index + key_index
results += uppercase[index % 26]
return results
```

You may have noticed that the signature for `vigenere_cipher()`

is quite similar to `caesar_cipher()`

from the previous section:

```
def vigenere_cipher(text, key, decrypt=False):
if not text.isascii() or not text.isalpha() or not text.isupper():
raise ValueError("Text must be uppercase ASCII without numbers.")
uppercase = string.ascii_uppercase
results = ""
```

The main difference is that, instead of a `shift`

parameter, `vigenere_cipher()`

takes a `key`

parameter, which is the keyword to be used during encryption and decryption. Another difference is the addition of `text.isupper()`

. Based on this implementation, `vigenere_cipher()`

can only accept input text that is all uppercase.

Like `caesar_cipher()`

, `vigenere_cipher()`

iterates over each letter of the input text to encrypt or decrypt it:

```
for i, char in enumerate(text):
current_key = key[i % len(key)]
```

In the above code, you can see the function’s first use of the modulo operator:

```
current_key = key[i % len(key)]
```

Here, the `current_key`

value is determined based on an index returned from `i % len(key)`

. This index is used to select a letter from the `key`

string, such as `M`

from `MODULO`

.

The modulo operator allows you to use any length keyword regardless of the length of the `text`

to be encrypted. Once the index `i`

, the index of the character currently being encrypted, equals the length of the keyword, it will start over from the beginning of the keyword.

For each letter of the input text, several steps determine how to encrypt or decrypt it:

- Determine the
`char_index`

based on the index of`char`

inside`uppercase`

. - Determine the
`key_index`

based on the index of`current_key`

inside`uppercase`

. - Use
`char_index`

and`key_index`

to get the index for the encrypted or decrypted character.

Take a look at these steps in the code below:

```
char_index = uppercase.index(char)
key_index = uppercase.index(current_key)
if decrypt:
index = char_index - key_index + 26
else:
index = char_index + key_index
```

You can see that the indices for decryption and encryption are calculated differently. That’s why `decrypt`

is used in this function. This way, you can use the function for both encryption and decryption.

After the `index`

is determined, you find the function’s second use of the modulo operator:

```
results += uppercase[index % 26]
```

`index % 26`

ensures that the `index`

of the character doesn’t exceed `25`

, thus making sure it stays inside the alphabet. With this index, the encrypted or decrypted character is selected from `uppercase`

and appended to `results`

.

Here’s the full code the Vigenère cipher again:

```
import string
def vigenere_cipher(text, key, decrypt=False):
if not text.isascii() or not text.isalpha() or not text.isupper():
raise ValueError("Text must be uppercase ASCII without numbers.")
uppercase = string.ascii_uppercase # "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
results = ""
for i, char in enumerate(text):
current_key = key[i % len(key)]
char_index = uppercase.index(char)
key_index = uppercase.index(current_key)
if decrypt:
index = char_index - key_index + 26
else:
index = char_index + key_index
results += uppercase[index % 26]
return results
```

Now go ahead and run it in the Python REPL:

```
>>> vigenere_cipher(text="REALPYTHON", key="MODULO")
DSDFAMFVRH
>>> encrypted = vigenere_cipher(text="REALPYTHON", key="MODULO")
>>> print(encrypted)
DSDFAMFVRH
>>> vigenere_cipher(encrypted, "MODULO", decrypt=True)
REALPYTHON
```

Nice! You now have a working Vigenère cipher for encrypting text strings.

## Python Modulo Operator Advanced Uses#

In this final section, you’ll take your modulo operator knowledge to the next level by using it with `decimal.Decimal`

. You’ll also look at how you can add `.__mod__()`

to your custom classes so they can be used with the modulo operator.

### Using the Python Modulo Operator With `decimal.Decimal`

#

Earlier in this tutorial, you saw how you can use the modulo operator with numeric types like `int`

and `float`

as well as with `math.fmod()`

. You can also use the modulo operator with `Decimal`

from the `decimal`

module. You use `decimal.Decimal`

when you want discrete control of the precision of floating-point arithmetic operations.

Here are some examples of using whole integers with `decimal.Decimal`

and the modulo operator:

```
>>> import decimal
>>> decimal.Decimal(15) % decimal.Decimal(4)
Decimal('3')
>>> decimal.Decimal(240) % decimal.Decimal(13)
Decimal('6')
```

Here are some floating-point numbers used with `decimal.Decimal`

and the modulo operator:

```
>>> decimal.Decimal("12.5") % decimal.Decimal("5.5")
Decimal('1.5')
>>> decimal.Decimal("13.3") % decimal.Decimal("1.1")
Decimal('0.1')
```

All modulo operations with `decimal.Decimal`

return the same results as other numeric types, except when one of the operands is negative. Unlike `int`

and `float`

, but like `math.fmod()`

, `decimal.Decimal`

uses the sign of the dividend for the results.

Take a look at the examples below comparing the results of using the modulo operator with standard `int`

and `float`

values and with `decimal.Decimal`

:

```
>>> -17 % 3
1 # Sign of the divisor
>>> decimal.Decimal(-17) % decimal.Decimal(3)
Decimal(-2) # Sign of the dividend
>>> 17 % -3
-1 # Sign of the divisor
>>> decimal.Decimal(17) % decimal.Decimal(-3)
Decimal("2") # Sign of dividend
>>> -13.3 % 1.1
1.0000000000000004 # Sign of the divisor
>>> decimal.Decimal("-13.3") % decimal.Decimal("1.1")
Decimal("-0.1") # Sign of the dividend
```

Compared with `math.fmod()`

, `decimal.Decimal`

will have the same sign, but the precision will be different:

```
>>> decimal.Decimal("-13.3") % decimal.Decimal("1.1")
Decimal("-0.1")
>>> math.fmod(-13.3, 1.1)
-0.09999999999999964
```

As you can see from the above examples, working with `decimal.Decimal`

and the modulo operator is similar to working with other numeric types. You just need to keep in mind how it determines the sign of the result when working with a negative operand.

In the next section, you’ll look at how you can override the modulo operator in your classes to customize its behavior.

### Using the Python Modulo Operator With Custom Classes#

The Python data model allows to you override the built-in methods in a Python object to customize its behavior. In this section, you’ll look at how to override `.__mod__()`

so that you can use the modulo operator with your own classes.

For this example, you’ll be working with a `Student`

class. This class will track the amount of time a student has studied. Here’s the initial `Student`

class:

```
class Student:
def __init__(self, name):
self.name = name
self.study_sessions = []
def add_study_sessions(self, sessions):
self.study_sessions += sessions
```

The `Student`

class is initialized with a `name`

parameter and starts with an empty list, `study_sessions`

, which will hold a list of integers representing minutes studied per session. There’s also `.add_study_sessions()`

, which takes a `sessions`

parameter that should be a list of study sessions to add to `study_sessions`

.

Now, if you remember from the converting units section above, `convert_minutes_to_day()`

used the Python modulo operator to convert `total_mins`

into days, hours, and minutes. You’ll now implement a modified version of that method to see how you can use your custom class with the modulo operator:

```
def total_study_time_in_hours(student, total_mins):
hours = total_mins // 60
minutes = total_mins % 60
print(f"{student.name} studied {hours} hours and {minutes} minutes")
```

You can use this function with the `Student`

class to display the total hours a `Student`

has studied. Combined with the `Student`

class above, the code will look like this:

```
class Student:
def __init__(self, name):
self.name = name
self.study_sessions = []
def add_study_sessions(self, sessions):
self.study_sessions += sessions
def total_study_time_in_hours(student, total_mins):
hours = total_mins // 60
minutes = total_mins % 60
print(f"{student.name} studied {hours} hours and {minutes} minutes")
```

If you load this module in the Python REPL, then you can use it like this:

```
>>> jane = Student("Jane")
>>> jane.add_study_sessions([120, 30, 56, 260, 130, 25, 75])
>>> total_mins = sum(jane.study_sessions)
>>> total_study_time_in_hours(jane, total_mins)
Jane studied 11 hours and 36 minutes
```

The above code prints out the total hours `jane`

studied. This version of the code works, but it requires the extra step of summing `study_sessions`

to get `total_mins`

before calling `total_study_time_in_hours()`

.

Here’s how you can modify the `Student`

class to simplify the code:

```
class Student:
def __init__(self, name):
self.name = name
self.study_sessions = []
def add_study_sessions(self, sessions):
self.study_sessions += sessions
def __mod__(self, other):
return sum(self.study_sessions) % other
def __floordiv__(self, other):
return sum(self.study_sessions) // other
```

By overriding `.__mod__()`

and `.__floordiv__()`

, you can use a `Student`

instance with the modulo operator. Calculating the `sum()`

of `study_sessions`

is included in the `Student`

class as well.

With these modifications, you can use a `Student`

instance directly in `total_study_time_in_hours()`

. As `total_mins`

is no longer needed, you can remove it:

```
def total_study_time_in_hours(student):
hours = student // 60
minutes = student % 60
print(f"{student.name} studied {hours} hours and {minutes} minutes")
```

Here’s the full code after modifications:

```
class Student:
def __init__(self, name):
self.name = name
self.study_sessions = []
def add_study_sessions(self, sessions):
self.study_sessions += sessions
def __mod__(self, other):
return sum(self.study_sessions) % other
def __floordiv__(self, other):
return sum(self.study_sessions) // other
def total_study_time_in_hours(student):
hours = student // 60
minutes = student % 60
print(f"{student.name} studied {hours} hours and {minutes} minutes")
```

Now, calling the code in the Python REPL, you can see it’s much more succinct:

```
>>> jane = Student("Jane")
>>> jane.add_study_sessions([120, 30, 56, 260, 130, 25, 75])
>>> total_study_time_in_hours(jane)
Jane studied 11 hours and 36 minutes
```

By overriding `.__mod__()`

, you allow your custom classes to behave more like Python’s built-in numeric types.

## Conclusion#

At first glance, the Python modulo operator may not grab your attention. Yet, as you’ve seen, there’s so much to this humble operator. From checking for even numbers to encrypting text with ciphers, you’ve seen many different uses for the modulo operator.

**In this tutorial, you’ve learned how to:**

- Use the
**modulo operator**with`int`

,`float`

,`math.fmod()`

,`divmod()`

, and`decimal.Decimal`

- Calculate the results of a
**modulo operation** - Solve
**real-world problems**using the modulo operator - Override
in your own classes to use them with the modulo operator`.__mod__()`

With the knowledge you’ve gained in this tutorial, you can now start using the modulo operator in your own code with great success. Happy Pythoning!