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# What is the value of positive integer x?

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Math Expert
Joined: 02 Sep 2009
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What is the value of positive integer x?  [#permalink]

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19 Jan 2015, 06:09
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46% (01:50) correct 54% (02:05) wrong based on 131 sessions

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What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

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Re: What is the value of positive integer x?  [#permalink]

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19 Jan 2015, 07:16
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What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer
Here is my solution though i am not sure that it is correct
Statement 1. 31 is out of scope since 31+1=32. Hence we should look for x<31. We can exclude all prime numbers (2,3,5,7,11,13,17,19,23,29 and 31) and multiples of 10 (10,20,30) so we have numbers such as 1,4,6,8,12,14,16,18,22,24,25,26,27,28. Lets exclude small numbers such as 1,4,6 since sum of their factors is less than 31 for sure. Lets check 12: 1,2,3,4,6,12=30 not enough. 14:1,2,7,14=24<31 16:1,2,4,8,16=31 ok. 25:1,5,25=31 ok. So we have two answers hence statement 1 is insufficient.
Statement 2. Not sufficient since x can be 9 or x can be 16
Both statements together. Still insuffient since x=16 and y=+4 or -4 or x=25 and y=+5 or -5. Hence answer E
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What is the value of positive integer x?  [#permalink]

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Updated on: 16 Feb 2015, 09:11
1
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

I think the answer is E:

My approach

from 1: sum of all unique factors of number x = 31. This is an odd number. so, the number could be a perfect square. trying out the perfect sqaured numbers below 31 we have:
1, 4, 9, 16, 25, 49 ..

sum of factors for 1: 1
sum of factors for 4: 7
sum of factors for 9: 13
sum of factors for 16: 1+2+4+8+16 = 31 -> possible number
sum of factors for 25: 1+5+25 = 31 --> possible number , already 2 possible solutions so NSF

from 2: given that x is a perfect sqaure, same info as from the stem --> NSF

combined we have 2 answers so NSF
E?

Originally posted by santorasantu on 19 Jan 2015, 12:56.
Last edited by santorasantu on 16 Feb 2015, 09:11, edited 2 times in total.
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Re: What is the value of positive integer x?  [#permalink]

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15 Feb 2015, 16:56
santorasantu wrote:
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

I think the answer is E:

My approach

from 1: sum of all unique factors of number x = 31. This is an odd number. so, the number should be a perfect square. trying out the perfect sqaured numbers we have:
1, 4, 9, 16, 25, 49 ..

sum of factors for 1: 1
sum of factors for 4: 7
sum of factors for 9: 13
sum of factors for 16: 1+2+4+8+16 = 31 -> possible number
sum of factors for 25: 1+5+25 = 31 --> possible number , already 2 possible solutions so NSF

from 2: given that x is a perfect sqaure, same info as from the stem --> NSF

combined we have 2 answers so NSF
E?

Though E is the correct answer, your analysis of statement 1 is faulty. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is not always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50.

From statement 2 we know that it is perfect square, which isn't sufficient. When we combine, we know it is a perfect square and it's factors add to 31, which gives us two answers as you've rightly mentioned.
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Re: What is the value of positive integer x?  [#permalink]

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16 Feb 2015, 09:09
aviram wrote:
santorasantu wrote:
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

I think the answer is E:

My approach

from 1: sum of all unique factors of number x = 31. This is an odd number. so, the number should be a perfect square. trying out the perfect sqaured numbers we have:
1, 4, 9, 16, 25, 49 ..

sum of factors for 1: 1
sum of factors for 4: 7
sum of factors for 9: 13
sum of factors for 16: 1+2+4+8+16 = 31 -> possible number
sum of factors for 25: 1+5+25 = 31 --> possible number , already 2 possible solutions so NSF

from 2: given that x is a perfect sqaure, same info as from the stem --> NSF

combined we have 2 answers so NSF
E?

Though E is the correct answer, your analysis of statement 1 is faulty. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is not always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50.

From statement 2 we know that it is perfect square, which isn't sufficient. When we combine, we know it is a perfect square and it's factors add to 31, which gives us two answers as you've rightly mentioned.

True you are correct, I purposefully picked only the perfect squares as to make the solution more concise.I edited my first statement.
Thanks for the point.
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Posts: 54440
Re: What is the value of positive integer x?  [#permalink]

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22 Feb 2015, 11:29
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Solution: E.

Statement 2 is clearly not sufficient as plenty of numbers are squares of integers (1, 4, 9, 16, 25, 36, etc.). When testing statement 1, then, it's a good idea to test candidates that will work with statement 2, as well. 25 is a good option: the factors of 25 are 1, 5, and 25 for a sum of 31. x could be 25. The only other number in that list with a chance is 16. And the factors of 16 are 1, 2, 4, 8, and 16. That sum is also 31. Because statements 1 and 2 each allow for two different values of x, the informations is not sufficient and the correct answer is E.
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Re: What is the value of positive integer x?  [#permalink]

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07 Sep 2017, 02:38
1) SUM OF ALL UNIQUE FACTORS = 31
X= 17x11x2x1 => sum =31
X= 17x13x1 => sum = 31
2) X=Y^2 , Y = integer
X=1 , Y=1
X=4, Y=2
X=9, Y=3
B eliminated

Combine 1&2

X= Y^2
Y= 17x13x1 => X= (17x13x1)^2
Y= 17x11x2x1 => X= (17x11x2x1)^2
C is eliminated

Ans is E
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Re: What is the value of positive integer x?   [#permalink] 07 Sep 2017, 02:38
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