Thread: Prime Power Congruence

Question:

Let a,b be distinct prime numbers.

Show that (a^(b-1) + b^(a-1) - 1) / (a*b) is an integer.

Should be easy but Im just completly blanked out...

Thank you

Originally Posted by Fulger85

Question:

Let a,b be distinct prime numbers.

Show that (a^(b-1) + b^(a-1) - 1) / (a*b) is an integer.

Should be easy but Im just completly blanked out...

Thank you

Hi Fulger85.

By Fermat’s little theorem, divides therefore divides

Similarly divides and so divides

Hence divides

Understood.

Thank you

Actually, do you mind elaborating on why lcm(a,b) divides it?

Thanks

Originally Posted by Fulger85

Actually, do you mind elaborating on why lcm(a,b) divides it?

Thanks

Property of lcm. If then , for .
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Your problem can be generalized to for relatively prime positive integers .