Proof If 7|(a^2+b^2) then 7|a and 7|b

[lukaszh]

Hello,
how to prove
If 7|(a^2+b^2) then 7|a and 7|b.
(If seven divides a^2+b^2 then seven divides a and seven divides b)
Thanks.

 

[tiny-tim]

Hello lukaszh!

(try using the X² tag just above the Reply box )

Hello,
how to prove
If 7|(a^2+b^2) then 7|a and 7|b.
(If seven divides a^2+b^2 then seven divides a and seven divides b)
Thanks.

Hint: what are 1², 2², 3² etc (mod 7)?

 

[lukaszh]

1²=1 (mod7)
2²=4 (mod7)
3²=2 (mod7)
4²=2 (mod7)
5²=4 (mod7)
6²=1 (mod7)
7²=0 (mod7)
Is it periodic {1,4,2,2,4,1,0} ? Now I know :-) It's periodic, so if I add any of these congruences together there will be some remainder. Remainder is zero if and only if I add congruences in form
(7k)²=0 (mod7)
(7j)²=0 (mod7)
THANX :-)