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QUESTION

Prove, by induction, that for all $n\geq1,\ 5^{2n}+7$ is divisible
by 8.



ANSWER

Take as inductive hypothesis the statement $8|5^{2n}+7$.

When $n=1,\ 5^{2n}+7=32=8.4$, so induction begins.

Now $8|5^{2n}+7$ implies $5^{2n}+7=8k$ for some integer $k$, so
$5^{2n}=8k-7$.

Thus $5^{2(n+1)}+7=5^{2n}.5^2+7=25(8k-7)+7=25.8k-7.24=8(25k-21)$.

Thus $5^{2(n+1)}+7$ is divisible by 8.

This completes the inductive step, so $8|5^{2n}+7$ is true for all
natural numbers $N$ by induction.




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