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Question 856598: Find the smallest natural number that can be expressed as the sum of the squares of two, not necessarily distinct, natural numbers in two different ways.
Show all work!
Thank you!
-Lulu
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! My answer is  , but my work is not that elegant.
I would post the problem in the forum of the artofproblemsolving website,
and then try to understand the very brief and elegant answer obtained there.
A pair of natural numbers  and  (not necessarily different from each other, maybe  ) squared ad up to  .
A different pair of natural numbers  and  (not necessarily different from each other) squared ad up to  .
One way would be making a table with
values for (from 1 to 10) in the first column,
values for (from 1 to 10) in the first row,
and the sum of squares where column meets row .
Comparing each values on each row, to other values
on the rows below, we would find many matching sums of squares of the same pair of numbers.
They are symmetrically arranged to either side of the diagonal.
We would be looking for matches between sums of squares of different pairs of numbers,
like  matching  ,
and  matching  .
Another approach:
--> -->
So we need an integer that can be written as two different products,
 and  .
 would lead you to solve systems like
 --> and
 -->
to get 
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