Paper 4, Section II, 6E6 \mathrm{E}

Numbers and Sets | Part IA, 2010

State and prove Fermat's Little Theorem.

Let pp be an odd prime. If p5p \neq 5, show that pp divides 10n110^{n}-1 for infinitely many natural numbers nn.

Hence show that pp divides infinitely many of the integers

5,55,555,5555,.5,55, \quad 555, \quad 5555, \quad \ldots .

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