Question

Prove. If p divides a and p divides a^2+b^2, then p divides b

Answer 1

do we know yet that:

a mod k + b mod k = (a+b) mod k ??

Answer 2

No

Answer 3

We are covering the gcd

Answer 4

I wa string to say that if p divides a and p divides a^2+b^2 the p must divide their difference but that doesnt seem to work

Answer 5

if not, then we simply have the definition of 'divides' to work with

p|a means that for some integer n: a = pn

p|(a^2+b^2) means that for some integer k: a^2+b^2 = pk

since a=pn, a^2 = p^2n^2 therefore

p^2n^2 + b^2 = pk

b^2 = pk - p^2n^2

can you work it from there?

Answer 6

since p divides the right side, p divides the left side should come into play with that

Answer 7

otherwise we might be able to work the quadratic in b

Answer 8

okay but i am not sure how that shows that p divides b if p divides b^2

Answer 9

bb = p (k-pn^2)

bb = p (k-an)

since integers are a closed set then k-an is also just some integer, let it be t

bb = pt

can we conclude from there? or do we need to go a different route ?

Answer 10

okay, that makes sense. thanks