Slide Deck: Modular Arithmetic

Slide 1: Clock Math

On a 12-hour clock, 13 lands where 1 lands.

That is the basic idea of modular arithmetic.

$13 \equiv 1 \pmod{12}$

We are not saying 13 equals 1. We are saying they have the same remainder after division by 12.

Slide 2: Remainders Are Positions

Working modulo $n$ means we only care about the remainder after division by $n$.

Examples:

• $17 \equiv 2 \pmod{5}$
• $24 \equiv 0 \pmod{6}$
• $31 \equiv 1 \pmod{10}$

Students should practice translating each one into a sentence.

Slide 3: Why This Helps

Large numbers become smaller.

Instead of calculating $7^{2026}$, we can look for the last digit pattern:

• $7^1 \equiv 7 \pmod{10}$
• $7^2 \equiv 9 \pmod{10}$
• $7^3 \equiv 3 \pmod{10}$
• $7^4 \equiv 1 \pmod{10}$

The cycle repeats every 4 powers.

Slide 4: Student Activity

Find the last digit of:

1. $3^{100}$
2. $2^{55}$
3. $9^{2026}$

Do not calculate the full number. Find the cycle.

Slide 5: Closing Idea

Modular arithmetic is not a trick for remainders.

It is a way of noticing that some information matters and some information can be safely ignored.