Prove, by mathematical induction, that \({7^{8n + 3}} + 2,{\text{ }}n \in \mathbb{N}\), is divisible by 5.

if \(n = 0\)

\({7^3} + 2 = 345\) which is divisible by 5, hence true for \(n = 0\)     A1

Note:     Award A0 for using \(n = 1\) but do not penalize further in question.

assume true for \(n = k\)     M1

Note:     Only award the M1 if truth is assumed.

so \({7^{8k + 3}} + 2 = 5p,{\text{ }}p \in  \bullet \)     A1

if \(n = k + 1\)

\({7^{8(k + 1) + 3}} + 2\)     M1

\( = {7^8}{7^{8k + 3}} + 2\)     M1

\( = {7^8}(5p - 2) + 2\)     A1

\( = {7^8}.5p - {2.7^8} + 2\)

\( = {7^8}.5p - 11\,529\,600\)

\( = 5({7^8}p - 2\,305\,920)\)     A1

hence if true for \(n = k\), then also true for \(n = k + 1\). Since true for \(n = 0\), then true for all \(n \in  \bullet \)     R1

Note:     Only award the R1 if the first two M1s have been awarded.

[8 marks]