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b divides a^n - 1, multiple of 7^{2010} of the form 99... 9

Source: 2010 Saudi Arabia Pre-TST 1.3

December 28, 2021

number theorydividesDigitsdivisible

Problem Statement

1) Let

a

and

b

be relatively prime positive integers. Prove that there is a positive integer

n

such that

1 \le n \le b

and

b

divides

a^n - 1

.
2) Prove that there is a multiple of

7^{2010}

of the form

99... 9

(

n

nines), for some positive integer

n

not exceeding

7^{2010}