24n+4 - 15n - 16, where n ∈ N is divisible by

Question

24n+4 - 15n - 16, where n ∈ N is divisible by (A) 225 (B) 227 (C) 229 (D) 285

Tardigrade Admin · Accepted Answer

We have, 24n+4 - 15n - 16 = 24(n+1) - 15n - 16 = 16n+1 - 15n - 16 = (1 + 15)n+1 - 15n - 16 = n+1C0 150 + n+1C2 151 + n+1C2 152 + n+1C3 153 + ... + n+1Cn+1 (15)n+1-15n-16 = 1+15n+15+ n+1C2 152+ n+1C3 153+... + n+1Cn+1 (15)n+1-15n-16 = 152[ n+1C2+ n+1C3 15+...+(15)n-1] Thus, 24n+4 - 15n - 16 is divisible by 225.

Q. $2^{4 n + 4} - 15 n - 16$ , where $n \in N$ is divisible by

$225$

$227$

$229$

$285$

Q. 24n+4−15n−16, where n∈N is divisible by

225

227

229

285

Q. $2^{4 n + 4} - 15 n - 16$ , where $n \in N$ is divisible by

$225$

$227$

$229$

$285$