mathschallenge.net

Problem

Consider the following results.

99¹ = 99
99² = 9801
99³ = 970299
99⁴ = 96059601
99⁵ = 9509900499

Prove that 99ⁿ ends in 99 for odd n.

Solution

Given that a b mod k, aⁿ bⁿ mod k.

As 99 1 mod 100, 99ⁿ (1)ⁿ mod 100

For odd n, (1)ⁿ = 1.

Therefore 99ⁿ 1 mod 100 for odd n; that is, one less than a multiple of 100, which means it will end in 99.

Prove that, for odd n, (i) 9ⁿ ends in 9, (ii) 999ⁿ ends in 999, (iii) Generalise.

If a b mod k, then prove that aⁿ bⁿ mod k.