For all n ∈ N, 23n+3 -7n- 8 is divisible by

Question

For all n ∈ N, 23n+3 -7n- 8 is divisible by (A) 49 (B) 17 (C) 343 (D) 81

Tardigrade Admin · Accepted Answer

23n + 3 - 7n- 8 = (23)n+1 -7(n + 1 ) - 1 = (1+7)n+1-7(n+1)-1 = n+1C0+ n+1C17+ n+1C2 72+ n+1C373+... + n+1Cn+1 7n+1 -7(n+1)-1 = 72( n+1C2+ n+1C371+... n+1Cn+1 7n-1) which is divisible by 49.

Q. For all $n \in N$ , $2^{3 n + 3} - 7 n - 8$ is divisible by

$49$

$17$

$343$

$81$

Q. For all n∈N, 23n+3−7n−8 is divisible by

49

17

343

81

23n+3−7n−8=(23)n+1−7(n+1)−1 =(1+7)n+1−7(n+1)−1 =n+1C0​+n+1C1​7+n+1C2​72+n+1C3​73+...+n+1Cn+1​7n+1 −7(n+1)−1 =72(n+1C2​+n+1C3​71+...n+1Cn+1​7n−1) which is divisible by 49.

Q. For all $n \in N$ , $2^{3 n + 3} - 7 n - 8$ is divisible by

$49$

$17$

$343$

$81$