A 4-digit number of the form aabb is a perfect square. What is the value of a - b?
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Answers
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.
7-4=3ans
Answer: A number of the form aabb has to be a multiple of 11.
So it is the square of either 11 or 22 or 33 or.... so on up to 99.
Step-by-step explanation:
88^2 = 7744
This is the only solution possible.
Therefore,
a = 7 and b = 4
Now,
a - b = 7 - 4 = 3
Hence, option (A) is correct. :)