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.

@TheSimpliFire Thanks, I just edited the question to be more specific. — Vivek Sivaramakrishnan, Sep 02 '18 at 08:48
Since you know that $a=b$ why don't you just write it as $aaaa$? — bof, Sep 02 '18 at 10:02

user · Accepted Answer · 2018-09-02T09:48:13.427

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

edited Sep 02 '18 at 09:48

answered Sep 02 '18 at 08:44

user

squares don't end in $3$, $7$ or $8$. – Angina Seng Sep 02 '18 at 09:37
@LordSharktheUnknown Yes of course! I fix that. Thanks – user Sep 02 '18 at 09:45

score 1 · Answer 2 · answered Sep 02 '18 at 08:46

1

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

answered Sep 02 '18 at 08:46

2 Answers2