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Let x x and y y be two 2-digit numbers such that y y is obtained by reversing the digits of x x . Suppose they also satisfy x 2 y 2 = m 2 x^{2}- y^{2} = m^{2} for some positive integer m m . Find x + y + m x + y + m .


Source: This question is from KVPY - 2014(SA) .


The answer is 154.

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2 solutions

Let x = a b , y = b a x = 10 a + b , y = 10 b + a x=\overline{ab},y=\overline{ba}\\x=10a+b,y=10b+a

Subbing in, we have

99 a 2 99 b 2 = m 2 99 m 1089 m Let m = 1089 k 2 , k N . 99 a 2 99 b 2 = 1089 k 2 a 2 b 2 = 11 k 2 ( a b ) ( a + b ) = 11 k 2 11 ( a b ) (rej) or 11 a + b Clearly, a + b < 20 a + b = 11 a b = k 2 99a^2-99b^2=m^2\\99|m\Rightarrow1089|m\\\mbox{Let }m=1089k^2,k\in\mathbb{N}.\\99a^2-99b^2=1089k^2\\a^2-b^2=11k^2\\(a-b)(a+b)=11k^2\\11|(a-b)\mbox{(rej)}\quad\mbox{or}\quad11|a+b\\\mbox{Clearly, }a+b<20\Rightarrow a+b=11\Rightarrow a-b=k^2

Setting k = 1 , 2 , 3 k=1,2,3 , we find that the unique solution is a = 6 , b = 5 a=6,b=5 . Hence, we have x + y + m = 65 + 56 + 33 = 154 x+y+m=65+56+33=154 .

Nicely presented,+1!!

Rishabh Tiwari - 4 years, 12 months ago
Vineet PaHurKar
Jun 16, 2016

Let x=10a+b Andy=10b+b Get x^2-y^2=m^2 That is 99(a+b)(a-b)=m^2. Here a and b are less than 10 positive integer. So a+b=11 and a-b =1 Hence x+y+m=154...

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