The largest $n$ for which $n+5$ divides $n^5+5$ is? $n$ being a natural number.
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Notice $(n+5)$ divides $n^5+5^5$(because $a+b$ always divides $a^n+b^n$ for odd $n$). So $n+5$ divides $n^5+5$ if and only if It divides $(n^5+5^5)-(n^5+5)=5^5-5=3120$.
The máximum is obtained by taking $n+5=3120$.
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can you elaborate please:) – srijan1791 Mar 04 '17 at 21:31
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I got the answer though.it is 3115. – srijan1791 Mar 04 '17 at 21:32
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I did it as follows: (n^5)+5=(n+5-5)^5+5. and then using binomial expansion you can come to the answer. – srijan1791 Mar 04 '17 at 21:32
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Elaborated ${}{}{}{}{}$ – Asinomás Mar 04 '17 at 21:40
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typo mate, its gotta be if and only if divides... – srijan1791 Mar 04 '17 at 21:45
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Why you wrote, "So n+5 divides $n^5+5$ if and only if It divides $(n^5+5^5)−(n^5+5)=5^5−5=3120$" ? – Iti Shree May 11 '17 at 13:48