Question

# A 4-digit number of the form aabb is a perfect square. What is the value of a - b?

CAT 2021

Correct option is

**Explanatory Answer :**

Number aabb can be written in expanded from as,

aabb = 1000a + 100a + 10b + b = 1100a + 11b = 11(100a + b)

For aabb to be a perfect square, 100a + b should be of the form 11n^2

, where n is a natural number.

∴ aabb = 11 × 11 × n^2

When n = 4,

11 × 11 × n^2

= 121 × 16 = 1936. This is not in the form aabb.

When n = 5,

11 × 11 × n^2

= 121 × 25 = 3025. This is not in the form aabb.

When n = 6,

11 × 11 × n^2

= 121 × 36 = 4356. This is not in the form aabb.

when n = 7,

11 × 11 × n^2

= 121 × 49 = 5629. This is not in the form aabb.

When n = 8,

11 × 11 × n^2

= 121 × 64 = 7744. This is in the form aabb.

When n = 9,

11 × 11 × n^2

= 121 × 81 = 9801. This is not in the form aabb.

So, 7744 is four digit number.

∴ a - b = 7 - 4 = 3

hence , A is the correct option