(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If a and b are integers and a is congruent to b(mod p) for every positive prime p, prove that a=b

2. Relevant equations

p divides (a-b) if a is congruent to b modulo p

if p divides ab then p divides a or p divides b (if p is prime)

3. The attempt at a solution

Suppose a is congruent to b(mod p)

so, p divides (a-b)

which means, there exists an integer c so that (a-b)=pc

where a=pc+b

(pc+b) is congruent to b(mod p)

so, p divides (pc+b-b)= (pc)

p divides (pc)

This is where i get stuck, i dont know if i should say since p is prime, p divides p or p divides c, or i don't know if i did this completely wrong. Any help would be appreciated =)

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# Homework Help: Congruence proof

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