Fermat, Congruences, the CRT, and an exam question

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July 12th 2011, 06:47 AM
jsndacruz

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 (Rofl)
July 12th 2011, 07:30 AM
Also sprach Zarathustra

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

Quote:

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 (Rofl)

From Fermat's theorem we get:

$p|q^{p-1}-1$ and $q|p^{q-1}-1$

Hence, $pq|(q^{p-1}-1)(p^{q-1}-1)$ .

So...

$0\equiv(q^{p-1}-1)(p^{q-1}-1)=...$

Try to continue.
July 13th 2011, 04:23 AM
chisigma

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

Quote:

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 (Rofl)

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

$q|(p^{q-1}-1)$

$p|(q^{p-1}-1)$ (1)

Now is...

$(p^{q-1}-1)\ (q^{p-1}-1) = p^{q-1}\ q^{p-1} - p^{q-1}- q^{p-1}+1$ (2)

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

$p\ q| (p^{q-1}+ q^{p-1}-1)$ (3)

Kind regards

$\chi$ $\sigma$
July 24th 2011, 07:49 PM
jsndacruz

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

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!
July 24th 2011, 08:17 PM
Also sprach Zarathustra

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

Quote:

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...