Simple Problem

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  Question: Find all two-digit positive integers such that the sum of the number and the number obtained by reversing the order of digits is a perfect square. Detailed solution: Let $xy$ be any such two-digit number (Note: Here $xy$ is just the form of a two-digit number like 25,43,etc., where $x$ and $y$ are digits). Therefore, $xy$ can be written as $10x+y$ (in expanded form). Number obtained by reversing the order of digits of $xy$ is $yx=10y+x$. According to question, The sum of $xy$ and the number obtained by reversing the order of digits of $xy$ i.e., $yx$ is a perfect square, which means $(10x+y)+(10y+x)=$Perfect square or, $11x+11y=$Perfect square or, $11(x+y)=$Perfect square .....$(1)$ But, maximum value of $x$ and $y$ are 9 each, so maximum value of $x+y$ is $9+9=18$. Observe that, to satisfy equation $(1), x+y=11k²$, for some positive integer $k$. For $k=1,x+y=11$ and for $k>1$ i.e., for $k≥2, x+y≥44$, which is not possible. Therefore, we get that $x+y=11$ is the only...
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SIMPLE QUESTION

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Question: Find all two-digit positive integers such that the sum of the number and the number obtained by reversing the order of digits is a perfect square.

Detailed solution: Let xy be any such two-digit number (Note: Here xy is just the form of a two-digit number like 25,43,etc., where x and y are digits)

Therefore, xy can be written as 10x+y (in expanded form). Number obtained by reversing the order of digits of xy is yx=10y+x

According to question, The sum of xy and the number obtained by reversing the order of digits of yx is a perfect square, which means

(10x+y)+(10y+x)=Perfect square

or, 11x+11y=Perfect square

or, 11(x+y)=Perfect square .....(1)

But, maximum value of x and y are 9 each, so maximum value of x+y is 9+9=18. Observe that, to satisfy equation (1), x+y=11k², for some positive integer k. For k=1,x+y=11 and for k>1 i.e., for k≥2, x+y≥44, which is not possible. Therefore, we get that x+y=11 is the only choice to satisfy equation (1).

Hence, (x,y)=(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2) satisfies. [Note: (0,11),(1,10),(10,1) and (11,0) are not possible because 10 and 11 are not digits]

So, all such two-digit positive integers are 29,38,47,56,65,74,83 and 92

Hope you all liked it!

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