Modular arithmetic

Finding the modulus

The modulus is the remainder when one number is divided by another.

It’s using represented using the modulus operator, often written as \% or mod.

Evaluate 7 \mod 4

To find 7 \mod 4, we divide 7 by 4:

• 7 \div 4 = 1 with a remainder of 3.
• Answer: 7 \mod 4 = 3.

Expressing modular binary operators

• We can write the modular arithmetic form of a binary operator using a subscript.
• For example, mod-5 addition would look like this:
  • a +_5 b

Modular addition

To find the modular addition of two numbers:

1. Add the two numbers together.
2. Find the modulus of the sum with respect to a given modulus.

Evaluate (5 + 9) \mod 6

• 5 + 9 = 14.
• 14 \mod 6 = 2 (since 14 \div 6 = 2 with a remainder of 2).
• Answer: 2

Evaluate (12 + 8) \mod 7

• 12 + 8 = 20.
• 20 \mod 7 = 6 (since 20 \div 7 = 2 with a remainder of 6).
• Answer: 6

Evaluate 15 +_4 10

• 15 + 10 = 25.
• 25 \mod 4 = 1 (since 25 \div 4 = 6 with a remainder of 1).
• Answer: 1

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