If p and x are integers , is x divisible by 11?

[BTGmoderatorAT]

If p and x are integers , is x divisible by 11?

(1) x = 2p - 6
(2) 2p + 5 is divisible by 11

What's the best way to determine whether statement 1 is sufficient? Can any experts help?

[Brent@GMATPrepNow]

If p and x are integers , is x divisible by 11?

(1) x = 2p - 6
(2) 2p + 5 is divisible by 11

Great question!!

Target question: Is x divisible by 11?

Given: p and x are integers

Statement 1: x = 2p - 6
This statement doesn't FEEL sufficient, so I'll TEST some values.
Case a: p = 10. So, x = 2(10) - 6 = 14. If x = 14, then the answer to the target question is NO; x is NOT divisible by 11
Case b: p = 14. So, x = 2(14) - 6 = 22. If x = 22, then the answer to the target question is YES; x IS divisible by 11
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of testing values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: 2p + 5 is divisible by 11
There is no information about x.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 2 tells us that 2p + 5 is divisible by 11
This means that 2p+5 = 11k for some integer k

Statement 1 tells us that x = 2p - 6
My goal is to fiddle with this equation so we can use the fact that 2p+5 = 11k
Notice that we can take the equation x = 2p - 6 and REWRITE it as x = 2p + 5 - 11 [since this new equation still simplifies to be x = 2p-6]
We can now replace 2p+5 with 11k to get: x = 11k - 11
We can now factor out 11 to get: x = 11(k - 1)
This tells us that x is a MULTIPLE of 11, which also means 11 is divisible by 11
So, the answer to the target question is YES; x IS divisible by 11
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent