True or False?

Algebra Level 3

If $N$ is divisible by $a+b$ , does it mean it is divisible by $a^m+b^m$ too?

True, only if m is odd. True, always. Cannot be determined. False. True, only if m is even.

1 solution

Sravanth C.
Oct 23, 2015

It is true but it only happens with odd integers. According to the question, $a+b\equiv 0 \pmod{N}\\a\equiv-b\pmod N$

If it has to be true for $a^m+b^m$ , $m$ has to odd, because if $m$ is even $-b^m$ will be positive. Instead $m$ needs to remain negative so that $N$ divides $a^m+b^m$ .

Note: Recall that if $a\equiv b\pmod N$ then, $a^m\equiv b^m\pmod N$ .

Moderator note:

Check the phrasing of your problem. I believe that the best answer is "False".

E.g. If 4 is divisible by $3 + 1$ , is it divisible by $3^m + 1^m$ ?

True or False?

1 solution

Moderator note:

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