Question

If $2^{4n+4}-15n-16,n\in N$ is divided by $225$, then the remainder is

• A. $0$
• B. $224$
• C. $1$
• D. $15$

Answer 1

The correct option is A $0$

Given:
$2^{4n+4}-15n-16=2^{4(n+1)}-15n-16={16}^{n+1}-15n-16=(1+15{)}^{n+1}-15n-16=1+{}^{n+1}{C}_1\cdot 15+{}^{n+1}{C}_2\cdot {15}^2+\dots {+}^{n+1}{C}_{n+1}\cdot (15{)}^{n+1}-15n-16=1+15(n+1)+{15}^2\left[k\right]-15n-16=225k,k\in N$

Hence, the remainder is $0$.