Show that $2^{4 n + 4} - 15 n - 16$ is divisible by 225

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Solution

$2^{4 n + 4} - 15 n - 16$

$= 2^{4 (n + 1)} - 15 n - 16$

$= 16^{n + 1} - 15 n - 16$

$(1 + 15)^{n + 1} - 15 n - 16$

$=^{n + 1} C_{0} 15^{0} +^{n + 1} C_{1} 15^{1} +^{n + 1} C_{2} 15^{2} +^{n + 1} C_{3} 15^{3} + . . . +^{n + 1} C_{n + 1} (15)^{n + 1} - 15 n - 16$

$= 1 + (n + 1) 15 +^{n + 1} C_{2} 15^{2} +^{n + 1} C_{3} 15^{3} + . . . +^{n + 1} C_{n + 1} (15)^{n + 1} - 15 n - 16$

$= 1 + 15 n + 15 +^{n + 1} C_{2} 15^{2} +^{n + 1} C_{3} 15^{3} + . . . +^{n + 1} C_{n + 1} (15)^{n + 1} - 15 n - 16$

$= 15^{2} [^{n + 1} C_{2} +^{n + 1} C_{3} 15 + . . .]$

Thus, $2^{4 x + 4} - 15 n - 16$ is divisible by $225$

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