## Finishing With 99

#### 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.

Problem ID: 141 (Dec 2003)     Difficulty: 4 Star

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