Question

What is 27^15 mod 33

Answer 1

First, 27 = 3³, so 27¹⁵ = (3³)¹⁵ = 3⁴⁵.

So we have

27¹⁵ ≡ 3⁴⁵ (mod 33)

and

3 ≡ 3 (mod 33)

3² ≡ 9 (mod 33)

3³ ≡ 27 (mod 33)

3⁴ ≡ 81 ≡ 15 (mod 33)

3⁵ ≡ 3 • 15 ≡ 45 ≡ 12 (mod 33)

From here, the cycle continues:

3⁶ ≡ 3 • 12 ≡ 36 ≡ 3 (mod 33)

3⁷ ≡ 3 • 3 ≡ 9 (mod 33)

and so on.

This is to say that, for each power of 3, where the power is a multiple of 5, the remainder will always be 12, i.e. for any positive integer k,

$3^{5k}\equiv 12\pmod{33}$