1. 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

2. 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, $a$ divides $b^{a-1}-1;$ therefore $a$ divides $a^{b-1}+b^{a-1}-1.$

Similarly $b$ divides $a^{b-1}-1$ and so $b$ divides $b^{a-1}+a^{b-1}-1.$

Hence $\mathrm{lcm}(a,b)=ab$ divides $a^{b-1}+b^{a-1}-1.$

3. Understood.

Thank you

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

Thanks

5. Originally Posted by Fulger85
Actually, do you mind elaborating on why lcm(a,b) divides it?

Thanks
Property of lcm. If $x|z,y|z$ then $\text{lcm}(x,y) |z$, for $x,y,z\in \mathbb{Z}^+$.
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Your problem can be generalized to $a^{\phi(b)} + b^{\phi(a)} \equiv 1(\bmod ab)$ for relatively prime positive integers $a,b$.