Question

Show that the number 9^{8n+4}-7^{8n+4} is divisible by 20 for any natural numbers n

Collected in the board: Number Theory

Steven Zheng posted 4 days ago

Answer

9^{8n+4}-7^{8n+4}

=81^{4n+2}-49^{4n+2}

=(81^{2n+1})^2-(49^{2n+1})^2

=(81^{2n+1}-49^{2n+1})(81^{2n+1}+49^{2n+1})

which shows the number could be factored to at least 2 factors.

The first factor is difference of two odd numbers which will result in a even number.

The second factor is the sum of 81^{2n+1}+49^{2n+1}

The ones place of 81^{2n+1} is 1. The ones place of 49^{2n+1} is 9. Their sum is 10

Therefore

9^{8n+4}-7^{8n+4} is divisible by both 2 and 10 simultaneously, hence, divisible by 20


Steven Zheng posted 4 days ago

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