Bunuel wrote:

The number 95 can be written as the sum of the squares of 4 different positive integers in two ways. Of the following, which can be the sum of these 4 integers?

(A) 9

(B) 10

(C) 15

(D) 18

(E) 20

Hi,

I will give you two ways..

1)

CHOICESEASY, if you understand properties of numbers and apply them correctly....

CRITICAL info - Since the SUM of squares of these 4 integers is ODD, the SUM of these numbers will also be ODD..ONLY 9 and 15 are ODD in choices...

The minimum SUM of 4 different positive integers = 1+2+3+4 = 10

so SUM>9....

ONLY 15 left

ans C

2) the biggest possible integer in these four integer will be 9 , as 10^2 goes above 95...

\(1^2 = 1.................2^2 = 4................3^2 = 9...................4^2 =16......................5^2 = 25...............6^2=36.............7^2= 49.............8^2=64..............9^2 = 81...\)

Since SUM is ODD, try combinations of 3 odd numbers and 1 even. OR 3 even and 1 odd......

now we can try some combinations and we will get our answer after some tries...

_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372

2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert