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The number 75 can be written as the sum

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by Brent@GMATPrepNow » Sat Feb 17, 2018 12:03 pm
Roland2rule wrote:The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

A) 17
B) 16
C) 15
D) 14
E) 13
We're looking for 3 DIFFERENT squares that add to 75

Here are the only squares we need to consider: 1, 4, 9, 16, 25, 36, 49, 64

Can you find 3 that add to 75?
After some fiddling, we may notice that 1 + 25 + 49
In other words, 1² + 5² + 7² = 75
We want the SUM of 1 + 5 + 7, which is 13

Answer: E

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by Scott@TargetTestPrep » Mon Jun 17, 2019 4:34 pm
BTGmoderatorRO wrote:The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

A) 17
B) 16
C) 15
D) 14
E) 13

If the sum of 3 different perfect squares is 75, each must be less than 75. So, we want to start by writing out the perfect squares that are less than 75:

1, 4, 9, 16, 25, 36, 49, 64

In this problem, there is no shortcut for determining which 3 squares add up to 75; however, it is strategic to start with the largest one, 64, and move down the list if it doesn't work:

64 + 11 = 75

There is no way 11 can be written as a sum of two squares. So, we move down to 49:

49 + 26 = 75

We see that 26 can be written as the sum of 25 and 1; that is:

49 + 25 + 1 = 75

We have found the three perfect squares that sum to 75. In other words:

7^2 + 5^2 + 1^2 = 75

The sum of 7, 5, and 1 is 7 + 5 + 1 = 13.

Answer: E

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