himu wrote: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