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

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

Binomial Theorem

Solution:

We have, $2^{4n+4} - 15n - 16 = 2^{4(n+1)} - 15n - 16$
$= 16^{n+1} - 15n - 16 = (1 + 15)^{n+1} - 15n - 16$
$= \,{}^{n+1}C_0\, 15^0 + \,{}^{n+1}C_2 \,15^1 + \,{}^{n+1}C_2\, 15^2 + \,{}^{n+1}C_3\, 15^3 + ...$
$+\,{}^{n+1}C_{n+1}\,\left(15\right)^{n+1}-15n-16$
$= 1+15n+15+\,{}^{n+1}C_{2}\,15^{2}+\,{}^{n+1}C_{3}\,15^{3}+...$
$+\,{}^{n+1}C_{n+1}\,\left(15\right)^{n+1}-15n-16$
$= 15^{2}\left[\,{}^{n+1}C_{2}+\,{}^{n+1}C_{3}\,15+...+\left(15\right)^{n-1}\right]$
Thus, $2^{4n+4} - 15n - 16$ is divisible by $225$.