if a and b are integers such that 3 a2 b2 then prove that

if a and b are integers such that 3 | (a^2 + b^2), then prove that 3 divides a and 3 divides b. Is the anologous statement true if 3 is replaced by 5?

Solution

Consider x^2 (mod 3) in three cases:

1) x = 3k;
2) x = 3k+1; and
3) x = 3k+ 2, where k is some integer.

1) x^2 = 9k^2 = 3*(3k^2) = 0 (mod 3).
2) x^2 = 9k^2 + 6k + 1 = 3*(3k^2 + 2) + 1 = 1 (mod 3).
3) x^2 = 9k^2 + 12k + 4 = 3*(3k^2 + 4k + 1) + 1 = 1 (mod 3).

Therefore, the sum a^2 + b^2 can be congruent to 0+0, 1+0, 0+1, and 1+1 (mod 3). The only case where 3 divides the sum is when the sum is congruent to 0 (mod 3), which is where they are congruent to 0+0 (mod 3). Thus, a^2 and b^2 are both congruent to 0 (mod 3), i.e. both are divisible by 3.