BTGModeratorVI wrote: ↑

Thu Dec 03, 2020 2:55 pm

If d and e are positive integers, is d a multiple of 9?

(1) 5e - 7 is a multiple of 9.

(2) d - 2 = 5e

Answer: C
Source: official guide

Target question: Is d a multiple of 9?

Statement 1: 5e - 7 is a multiple of 9
This statement provides no information about d
Statement 1 is NOT SUFFICIENT

Statement 2: d - 2 = 5e
It's easier to analyze the statement if we rewrite the equation as: d = 5e + 2
There are several values of d and e that satisfy this equation. Here are two:
Case a: d = 7 and e = 1. Since 7 is not a multiple of 9, the answer to the target question is NO, d is not a multiple of 9
Case b: d = 27 and e = 5. Since 27 is a multiple of 9, the answer to the target question is YES, d is a multiple of 9
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that 5e - 7 is a multiple of 9.
This means we can write: 5e - 7 = 9k (for some integer value of k)

Statement 2 tells us that d - 2 = 5e, which we can rewrite as: d = 5e + 2

KEY STEP: Take d = 5e + 2 and rewrite it as: d = 5e - 7 + 9

Aside: I did this because we have some very specific information about 5e - 7 (it equals 9k, for some integer value of k)

So, we have: d = 5e - 7 + 9
Which is the same as: d = (5e - 7) + 9
Now substitute to get: d = 9k + 9
Now factor out the 9 to get: d = 9(k + 1)
At this point, it's clear that d is a multiple of 9
The answer to the target question is YES, d is a multiple of 9
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent