Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Novemberish

Age 14 to 16 Challenge Level:

(a) A four digit number (in base $10$) $aabb$ is a perfect
square. Discuss ways of systematically finding this number.
(b) Prove that $11^{10}-1$ is divisible by $100$.