SOLUTION: Two numbers between 150 and 200 can each be expressed as the sum of two squares in two different ways. Find their difference. CC13F #2

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Question 1209470: Two numbers between 150 and 200 can each be expressed as the sum of two squares in two different ways. Find their difference.
CC13F #2
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52775)   (Show Source): You can put this solution on YOUR website!
.
Two numbers between 150 and 200 can each be expressed as the sum of two squares in two different ways.
~~~~~~~~~~~~~~~~~ 

Make the table of squares of natural numbers between 0 and 200  (not so many opportunities)

    0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196.


Make the table of sums of squares between 150 and 200  (not so many opportunities)

 100 + 64 = 164
 121 + 25 = 146
 121 + 36 = 157
 121 + 49 = 170
 121 + 64 = 185  (**)
 121 + 81 = 202
 144 + 25 = 169   (*)
 144 + 36 = 180
 144 + 49 = 193
 169 +  0 = 169   (*)
 169 +  9 = 178
 169 + 16 = 185  (**)
 169 + 25 = 194
 196 +  0 = 196


The repeating sums in this list are  

    169 = 169 + 0 = 144 + 25   (marked * in the list),

and

    185 = 121 + 64 = 169 + 16  (marked ** in the list).


So, these two numbers, 169 and 185 are those you are looking for.


Their difference is  185 - 169 = 16.    ANSWER

Solved.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

This is something to find through trial-and-error.
Or you can use technology.

Of the integers in the interval 150 < x < 200, only the values 170 and 185 have two distinct ways to break them down into a sum of two squares
170 = 1+169 = 49+121
185 = 16+169 = 64+121
I'll assume that 0 isn't being considered.

Therefore 185-170 = 15 is the final answer.
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