The Unique Square number !

Number Theory Level 2

Which is the only 2 2 digit number whose square can be written in the form of X X Y Y XXYY ?


The answer is 88.

Shreya R by Shreya R , 21, India

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4 solutions

X X Y Y = 1100 X + 11 Y = 11 ( 100 X + Y ) XXYY = 1100X + 11Y = 11(100X + Y)

Note: X and Y lie in [ 1 , 9 ] [1,9] .

Since it's a square number and has 11 as a factor, 100 X + Y 100X + Y has 11 as a factor too. The smallest number in this format AND has 11 as a factor is 209, by observation. the hundreds digit increments and the units digit decrements, so the following numbers are possible solutions: 209, 308, 407, ... , 902.

By trial the solution is 88 \boxed{88} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Asher Joy
Jun 4, 2014

88^2 = 7744 (XXYY), so therefore 88 is the only two digit number with this property.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

is there any technical method????

Mayank Holmes - 7 years ago

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Yes there is. Write down the number in decimal exapnsion and maybe try. Sorry, i am too lazy to use latex

Krishna Ar - 7 years ago

If it is in the form XXYY, then it is a multiple of 11, so you only have to test 9 of them.

Frodo Baggins - 7 years ago
Ishma Narag
Jun 6, 2014

Ummm, 88 and -88 are two 2-digit numbers whose square is in the form XXYY = 7744.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

So answer should be both he numbers

Akhil Krishna - 5 years, 7 months ago

Do trial and error method! Square digits if you can!

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