Problem Solving

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

Please Explain....

the number 75 is the sum of squares of three numbers...... i solved it in trail and eror method....... 7+5+1=13 (i.e 49+25+1=75)

siddarthd2919 wrote:the number 75 is the sum of squares of three numbers...... i solved it in trail and eror method....... 7+5+1=13 (i.e 49+25+1=75)

Trial and error is almost certainly the quickest way to answer this question.

Write out perfect squares under 75:

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

Find 3 different numbers on the list that sum to 75, add up their roots.

"Different" is a key word, otherwise 25 + 25 + 25 would also have been a valid solution.

Given
x^2+y^2+z^2=75………………………… (1)
Suppose,
x+y+z = k…………………………………(2)
Now, we know from the standard formula

x^2+y^2+z^2 +2(xy+yz+zx) = (x+y+z)^2

Putting the values from (1) & (2)

=>75+2(xy+yz+zx) = k^2
=> xy+yz+zx = (k^2 -75)/2 ………..(3)

Since, x,y and z are all integers, therefore, (k^2 -75)/2 must be an integer,

So, (k^2 -75) must be an even integer
For (k^2 -75) to be even, k^2 must be odd, ( since, odd – odd = even)
For k^2 to be odd, k must be odd

Thus only k = 13, 15 and 17 can be the sum of integers,

Putting the value in (3), xy+yz+zx = 47,75,107

But from the inequality, xy+yz+zx < x^2+y^2+z^2

Only value of xy+yz+zx satisfying this is 47

Thus, k= x+y+z = 13…………………………………………………….Ans


Hope it is clear to you.

I really don't see how this question can be answered within 2 mins even by using trial and error. Can someone explain how it can be solved this way within a reasonable amount of time?

Thanks!

Easier if you just say 75=x^2+ y^2 + z^2
then 75= 25+50
25 is the perfect square of 5 so you have x.
50= 49 +1 ; 49 is the perfect square of 7 so you have y and 1 is square of 1 so z.
x+y+z=13

I don't know if it's correct, anyway:

75=5x5x3
5+5+3=13

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

75=a^2 + b^2 + c^2
75=25 + 49 + 1 = 5^2 + 7^2 + 1^2
a+B+c = 5+7+1 = 13

Here is how I worked it. I broke down 75 into its prime factors ie
75 = 5 x 5 x 3
I saw that the prime factors made up one perfect square I could see immediately ie
5 x 5 = 25 ------ So I figured 25 was one of the perfect squares
I subtracted 25 from 75 and was left with 50
This meant x-squared + y-squared MUST equal 50
ALSO I could see that both x-squared and y-squared MUST be odd (odd/even properties)
This enabled me to elminate for consideration any even perfect squares under 50
SO I was left with ONE, NINE and FORTY-NINE as the only possibilities.
The rest is a peice of cake

*IF the answer is not obvious by this time then you could do this
(1 + 9 + 49)-50 = 9 (this means the odd man out is nine and 1 and 49 remain as your perfect squares!) Consider taking the GRE if you have to do this last step !!!!!!(ha ha that was mean)