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Question

Prove that 52n+124n25 is divisible by 576.

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Solution

Let 52n+224n25 be denoted by f(n);
then f(n+1)=52n+424(n+1)25
=5252n+224n49;
f(n+1)25f(n)=25(24n25)24n49
=576(n+1).
Therefore if f(n) is divisible by 576, so also is f(n+1); but by trial we see that the theorem is true when n=1, therefore it is true when n=2, therefore it is true when n=3, and so on; thus it is true universally.
The above result may also be proved as follows:
52n+224n25=25n+224n25
=25(1+24)n24n25
=25+25.n.24+M(242)24n25
=576n+M(576)
=M(576).

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