Question

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

A. 3
B. 2
C. 4
D. 1

Answer 1

A. 3

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Since aabb is a 4-digit number, first we determine limitations:

• 0 < a ≤ 9 and
• 0 ≤ b ≤ 9

Next, lets convert the number as:

• aabb = aa00 + bb = 1100a + 11b = 11(100a + b)

Since aabb is a perfect square and it has a factor of 11, we need 100a + b to have a factor of 11:

• 100a + b = 99a + (a + b)

Here 99a is divisible by 11 and a + b must be divisible by 11.

It is possible if:

• 1) a + b = 0, which is impossible due to limitations;
• 2) a + b = 11

We need to consider that no perfect square ends with 2, 3, 7, 8, hence:

• b ≠ 2, 3, 7, 8

It means:

• a ≠ 9, 8, 4, 3 and a ≠ 1 since b < 10

So all the possible values of a:

• a = 7, 6, 5, 2

Lets try the possible pairs of a and b:

• 1) a = 7, b = 4 ⇒ aabb = 7744, this is a perfect square
• 2) a = 6, b = 5 ⇒ aabb = 6655, this is not a perfect square
• 3) a = 5, b = 6 ⇒ aabb = 5566, this is not a perfect square
• 4) a = 2, b = 9 ⇒ aabb = 2299, this is not a perfect square

Hence the value of a - b is:

• a - b = 7 - 4 = 3

The matching choice is A.