If positive integer y is equal to the sum of all the unique

Question

If positive integer y is equal to the sum of all the unique factors of the positive integer x, is |x-y| > 1?

(1) x is not prime.
(2) x does not equal 1.

KarishmaB · Accepted Answer

So here is the thing about sum of unique factors of positive integers:
Except 1 every positive integer has 2 or more factors - 
Prime numbers have exactly 2 factors: 1 and itself - 
Composite numbers have more than 2 factors: 1, itself and at least one more.

So if x = 1, sum of factors (y) = 1
If x = prime, sum of factors (y) = x+1
If x is Composite, sum of factors (y) = x+1+ (At least one more factor)

So difference between x and y is greater than 1 for only the composite numbers.

Statement 1 tells us that x is not prime and statement 2 tells us that it is not equal to 1 (1 is neither prime nor composite). So together, we know that x is composite. Hence we need both statements to answer the question.

Answer (C)

cherukuri1011 · Answer

The answer is C

Positive integer y is equal to the sum of all the unique factors of the positive integer x
1. X is not prime.
if X is not prime then |x-y| >1 except for 1
for example if x=4 then y=1+2+4 =7
for x=1 y=1
so insufficient
2. X does not equal to 1
If x = prime =3
y =1+3 (prime number factors are 1 and itself)
so,|x-y| = 1
hence, Insufficient
if we combine 1 and 2 then x is not prime and x not equal to 1 then |x-y| > 1 always.
hope this helps.
Give me kudos if this clarifies

AKG1593 · Answer

Statement 1:X is not prime
Implies that the absolute difference will always be greater than 1 except for just one case when x=1.Insufficient
Statement 2 in itself is also insufficient.When x=2,y=3 & absolute diff=1(not greater than 1)
It will be greater than 1 for x=4 etc.
Combining both statements, we have x=neither prime nor 1,so absolute diff of x & y will always be greater than 1.
Ans C

Posted from my mobile device

EMPOWERgmatRichC · Answer

Hi All,

This question is built around several Number Properties, but you don't need to know those rules to get the question correct. TESTing VALUES and pattern-matching will get you the correct answer.

We're told that X and Y are positive integers and the Y = the sum of all the unique factors of X. We're asked if |X-Y| > 1. This is a YES/NO question.

Fact 1: X is NOT prime

IF....
X = 1
The only factor of 1 is 1
Y = 1
|1-1| = 0 and the answer to the question is NO

IF....
X = 4
The factors of 4 are 1, 2 and 4
Y = 7
|4-7| = 3 and the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: X does NOT = 1

IF....
X = 2
The factors of 2 are 1 and 2
Y = 3
|2-3| = 1 and the answer to the question is NO

IF....
X = 4
The factors of 4 are 1, 2 and 4
Y = 7
|4-7| = 3 and the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know....
X is NOT prime
X is NOT 1

IF....
X = 4
The factors of 4 are 1, 2 and 4
Y = 7
|4-7| = 3 and the answer to the question is YES

IF....
X = 6
The factors of 6 are 1, 2, 3 and 6
Y = 12
|6-12| = 6 and the answer to the question is YES

From these examples, you can see that since X is NOT prime (and NOT 1), then it will have factors BESIDES itself and the number 1. Thus Y will always be MORE than 1 greater than X and the answer to the question will ALWAYS be YES.
Combined, SUFFICIENT

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

schazamhuzzah · Answer

Hi Karishma, what about perfect numbers?

If we take the example of 6(from the given conditions its not prime and not 1) : factors of 6 are 1,2,3 sum=1+2+3 is 6 ..here x-y is 0.

Can you pls explain this ...I marked the answer as E based on this reasoning

EMPOWERgmatRichC · Answer

Hi schazamhuzzah,

You've made a minor mistake with your example:

The factors of 6 are 1, 2, 3 AND 6.....so the sum of the factors is 12 (not 6). By extension, the value of |X-Y| = |6-12| = |-6| = 6.

GMAT assassins aren't born, they're made,
Rich

ggurface · Answer

I got tricked and answered A because I thought that 1 was a prime.

For |x-y| > 1, x cant be 1, 2, 3, 5, 7, any prime

For x = 1, |x-y| = 0. For x = 2 and 3, their only factors are themselves and 1, so |x-y| = 1 which is not > 1. Because I thought 1 was a prime, I thought it was eliminated from the possible result set, so I thought it was sufficient because it was telling us x = 4 or more.

In reality, since 1 is NOT a prime, statement 1 still gives 2 possible answers |x-y| < 1 if x = 1, and > 1 if x = 4, 6, etc.

Therefore statement 2 completes for us because it elimiantes 1, and x is 4, 6, 8, etc.

Note to self, memorize properties of numbers for edge cases near 0!

KarishmaB · Answer

Every number has at least 2 factors: 1 and itself. You forgot the itself part. It may have other factors too. So 6 has 6 as a factor too. For more on factors, check this post: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/12 ... ly-number/

BillyZ · Answer

Official solution from Veritas Prep.

Solution: C

Try to break the pattern first. If x is prime, it will have two factors: itself and 1, so  y = x+1 .  |x-(x+1)| = 1 , so if x is prime  |x-y|  will  not  be  greater than 1 . If x is exactly one, then y = x and  |x-x| = 0 , which is not greater than 1. If x is ANY OTHER eligible integer (a positive integer greater than 1 that is not prime)  |x-y| > 1 . (Try a few if you like – the extra factors other than the number itself and 1 guarantee that  y  is at least  = x+2 .) Statement (1) is thus not sufficient, nor is statement (2), but together both primes and 1 are ruled out, meaning that  |x-y| > 1 . (C).

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If positive integer y is equal to the sum of all the unique

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If positive integer y is equal to the sum of all the unique

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards