Given $aabb$ is a square number, and $a := b$, find $a$ and $b$.

Question

I want to solve the above question systematically, i.e, assuming that I do not know all the $4$-digit square numbers.

Answer 1

We have that

$$aabb=1100a+11b=11(100a+b)$$

then we need

$$11|100a+b \iff a+b\equiv0 \pmod{11}$$

moreover

$$1100a+11b\equiv 0,1 \pmod 4 \iff 3b \equiv 0,1 \pmod 4 \iff b \equiv 0,3 \pmod 4$$

but since squares doesn't end with $3$, $7$ or $8$ then we need to check among

• $0000,7744$

Answer 2

Hint: Remember that your number can be written in the form $$z=b+10b+100a+1000a=11b+1100a$$