Hi people! That's my first time here in this forum, and I want to share with you some doubts.

My teacher of number's theorie suggested about 40 exercises about congruences, and I'm having trouble in some (5) of them.

There are the exercises:

EDIT: NOTATION: (a,b) is the greatest common divisor of a and b; N for Naturals, Z for Integers and P for primes; $\displaystyle \phi (a)$ is the Euler Totient of a.

1st)If $\displaystyle p, q \in P$ and $\displaystyle p \neq q$, so $\displaystyle p^{q-1} + q^{p-1} \equiv 1 \pmod{p.q}$.

-Observation: I think one problem is that I don't know how to create an useful relation in mod pq using relations on mod p and mod q. The only way that I know is the way that goes with the property: If $\displaystyle a \equiv b (mod n)$, so $\displaystyle a.c \equiv b.c \pmod{c.m}$ .

2nd) This second exercise is an generalization of the previous one (they are presented in this order).

If $\displaystyle a, b \in N$ and $\displaystyle (a,b) = 1$ , so $\displaystyle a^{\phi (b)} + b^{\phi (a)} \equiv 1 \pmod{a.b}$

3rd) The third problem is how to proof that$\displaystyle 42|a^7 - a, \forall a \in Z. $

4th) Exercise:If $\displaystyle p, q \in P$, with $\displaystyle p \neq q$, are related as follows: $\displaystyle a^p \equiv a \pmod{q}$ and $\displaystyle a^q \equiv a \pmod{p}$, so $\displaystyle a^{pq} \equiv a \pmod{pq}$.

-Obs.: I think the observation posted in the first exercise will affect this one.

5th)If $\displaystyle a \in Z$ and $\displaystyle p,q \in P$, with $\displaystyle p \neq q$, so $\displaystyle p.q | a^{p+q} - a^{p+1} - a^{q+1} + a^2$.

I hope you can help me, any help (or try to) is very welcome!

Thanks,

Rodrigo.