Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
[Math] Find the four digit number
contest-mathelementary-number-theoryrecreational-mathematics
contest-mathelementary-number-theoryrecreational-mathematics
Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
Best Answer
HINT:
So, we have $$1000a+100a+10b+b=11(100a+b)$$
$\implies 100a+b$ must be divisible by $11\implies 11|(a+b)$ as $100\equiv1\pmod{99}$
As $0\le a,b\le 9, 0\le a+b\le 18\implies a+b=11$
$$\implies11(100a+b)=11(100a+11-a)=11^2(9a+1)$$
So, $9a+1$ must be perfect square