Thread: Fermat, Congruences, the CRT, and an exam question

Had the following problem on an exam, and couldn't crack it:

Show that if p and q are primes, then p^(q-1) + q^(p-1) == 1 (mod pq)

We know through Fermat each p^(q-1) == 1 (mod q) and q^(p-1) == 1(mod p). However I'm not sure how to combine the two.

I thought the CRT might help if we changed the system to p^(q) == p(mod q) and q^(p) == q (mod p).

If you think the problem easy, feel free to leave a hint rather than a solution so I have to work a bit -- I'd appreciate either

Originally Posted by jsndacruz

Had the following problem on an exam, and couldn't crack it:

Show that if p and q are primes, then p^(q-1) + q^(p-1) == 1 (mod pq)

We know through Fermat each p^(q-1) == 1 (mod q) and q^(p-1) == 1(mod p). However I'm not sure how to combine the two.

I thought the CRT might help if we changed the system to p^(q) == p(mod q) and q^(p) == q (mod p).

If you think the problem easy, feel free to leave a hint rather than a solution so I have to work a bit -- I'd appreciate either

From Fermat's theorem we get:

and

Hence, .

So...



Try to continue.

Originally Posted by jsndacruz

Had the following problem on an exam, and couldn't crack it:

Show that if p and q are primes, then p^(q-1) + q^(p-1) == 1 (mod pq)

We know through Fermat each p^(q-1) == 1 (mod q) and q^(p-1) == 1(mod p). However I'm not sure how to combine the two.

I thought the CRT might help if we changed the system to p^(q) == p(mod q) and q^(p) == q (mod p).

If you think the problem easy, feel free to leave a hint rather than a solution so I have to work a bit -- I'd appreciate either

For the 'Fermat little theorem', if p and q are primes then...



(1)

Now is...

(2)

... and from (1) and (2) You easily derive that...

(3)

Kind regards

Thanks so much, also sprach Zarathustra (didn't know that was the song's name until my latest google of your name) and ChiSigma. Sorry for not thanking earlier!

Originally Posted by jsndacruz

Thanks so much, also sprach Zarathustra (didn't know that was the song's name until my latest google of your name) and ChiSigma. Sorry for not thanking earlier!

Now you just left to read Nietzsche...