Modular Arithmetic

In modular arithmetic, numbers "wrap around" when reaching a certain value, called the "modulus". The calculation is simple.

a = b (mod n)

Using the modulo operator %, we first do a % n = b. Now we know number a is congruent with the remainder, b, in (mod n). Likewise, all numbers with this remainder are congruent in (mod n).

(Mod 2)

In the world of (mod 2), 0 is congruent with 0 (mod 2). So we would write:

0 ≡ 0 (mod 2)

And so on:

• 0 ≡ 0 (mod 2)
• 1 ≡ 1 (mod 2)
• 2 ≡ 0 (mod 2) wraps back to zero!
• 3 ≡ 1 (mod 2)
• 4 ≡ 0 (mod 2) wraps back to zero!
• ...

(Mod 3)

As you saw above, in the world of (mod 2), there are only 2 possible values: {0, 1}. However, in the world of (mod 3), there are 3 possible values: {0, 1, 2}.

• 0 ≡ 0 (mod 3)
• 1 ≡ 1 (mod 3)
• 2 ≡ 2 (mod 3)
• 3 ≡ 0 (mod 3) wraps back to zero!
• 4 ≡ 1 (mod 3)
• 5 ≡ 2 (mod 3)
• 6 ≡ 0 (mod 3) wraps back to zero!
• ...