Question

In a 4-digit perfect square, the first two digits are the same, and the last two digits are also the same. What is the value of this 4-digit number?

Answer 1

• 7744

Explanation:

Let the number is in the form of aabb.

We can put it as:

• aabb = 11*(100a + b) = 11*(99a + a + b)

The number is a perfect square so it must be divisible by 11.

It is divisible by 11 if (a + b) is divisible by 11..

On the other hand, b = 0, 1, 4, 5, 6, 9 as the last digit of a perfect square.

Also, both a and b must be within (0,9) interval.

Considering the above conditions we have options:

• a,b = 2,9 or 5,6 or 6,5 or 7,4

The numbers are:

• 2299
• 5566
• 6655
• 7744

By testing we confirm only one of them is a perfect square:

• 7744