Does 11 has some use here?
Number Theory
Level
2
Find a 4 digit perfect square number,such that it's first 2 digits and the last 2 digits are the same respectively.
The answer is 7744.
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1 solution
Let the number be xxyy.
xxyy = 1000x+100x+10y+y = 1100x+11y = 11(100x+y)
So the number is multiple of 11. To be a perfect square, it should be a multiple of 112 ie.,121 The other multiplicand should also be a perfect square.
Now we will try out some possibilities:
121 * 22 = 484 => option is out as its not a 4 digit number
121 * 32 = 1089 => option is out as first and second digits are not same as is the case for third and fourth digits also.
121 * 42 = 1936 => option is out as first and second digits are not same as is the case for third and fourth digits also.
121 * 52 = 3025 => option is out as first and second digits are not same as is the case for third and fourth digits also.
121 * 62 = 4356 => option is out as first and second digits are not same as is the case for third and fourth digits also.
121 * 72 = 5929 => option is out as first and second digits are not same as is the case for third and fourth digits also.
121 * 82 = 7744 => This suits