is |x-y| > 1?

[himu]

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

x is not prime
x≠1

[aditya8062]

my take wud be C .plz confirm so as i may put solution

[srcc25anu]

Y = sum of unique factors of x
is |x-y| > 1?

St1: x <> Prime

if x = 1, y = 1 and |x-y|= 0 (NOT GREATER THAN 1)
if x = 4, y = 7 and |x-y|= 3 (GREATER THAN 1)
Not sufficient

St2: x <> 1
if x = 2, y = 3 and |x-y|= 1 (NOT GREATER THAN 1)
if x = 4, y = 7 and |x-y|= 3 (GREATER THAN 1)
Not sufficient

ST1 + ST2 together:
x = 4, y = 7 and |x-y|= 3 (GREATER THAN 1)
x = 6, y = 12 and |x-y|= 6 (GREATER THAN 1)
x = 8, y = 15 and |x-y|= 7 (GREATER THAN 1)

as x increases, y will increase at a faster rate such that |x-y| will always be greater than 1

hence C. Both statements are sufficient

[Brent@GMATPrepNow]

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

x is not prime
x≠1

Target question: Is |x-y| > 1?

Given: positive integer y is equal to the sum of all the unique factors of the positive integer x

There are 3 cases to consider here.

case a: x = 1
This means y = 1
In this case, |x-y| = 0

case b: x is prime
Here, the only factors of x are: 1 and x, which means y = x+1
In this case, |x-y| = |x-(x+1)| = |-1| = 1

case c: x is composite
Here, the factors of x are: 1, x, and at least one extra number, which means y = x+1+(at least one extra number)
In this case, |x-y| > 1

Statement 1: x is not prime
This rules out case b, which leave us with case a or case c
In case a: |x-y| = 0
In case c: |x-y| > 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x≠1
This rules out case a, which leave us with case b or case c
In case b: |x-y| = 1
In case c: |x-y| > 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Statement 1 rules out case b
Statement 2 rules out case a
This leaves us with case c, which means |x-y| > 1
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

[GMATGuruNY]

I also posted an explanation here:

https://www.beatthegmat.com/if-positive- ... tml#619012