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

(1) 5e - 7 is a multiple of 9. - Insufficient - as no info for d.

(2) d - 2 = 5e - Insufficient - as it may or may not be .... depending on values of d and e.
e.g., when e=1, d=7 -----> d is not a multiple of 9
but when e=5, d= 29 -----> d is a multiple of 9

Combining (1+2) ==> say , 5e - 7 = 9x ==> 5e=9x+7
hence , d-2 =9x+7 ==>d=9x+9 ==>d=9(x+1) ===> d definitely a multiple of 9 -Sufficient ...............Ans C

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

BrentGMATPrepNow when we look at (2) or any statement like statement (2) do we need to find a few values to get a Yes and No situation OR could we simply see that since we have d = 5e + 2 i.e. since we have two variables and one constant we WILL get different values?

I know that on the GMAT we can have an equation with 2 variables that can be sufficient BUT in such questions we are given restrictions that help us get one definite answer. But in this question we do not have any restriction about d or e. So could we just eliminate (2) without analyzing it much?

It's difficult to derive a clear-cut rule for when you need to actually find values that show a statement is not sufficient.

For example, if statement 2 were something like d - 27 = 18e, then statement 2 would be sufficient.
Likewise, if statement 2 were something like d - 26 = 18e, then statement 2 would also be sufficient.

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

(1) 5e - 7 is a multiple of 9.
Nothing is mentioned about d
NOT SUFFICIENT

(2) d - 2 = 5e
d = 5e + 2
If d=27; d is a multiple of 9
But if d=22; d is NOT a multiple of 9.
NOT SUFFICIENT

(1) + (2)
(1) 5e - 7 is a multiple of 9.
(2) d - 2 = 5e
d = 5e + 2 = (5e-7) + 9 = 9k + 9 ; d is a multiple of 9
SUFFICIENT

IMO C

S1 doesnt give us any information about d, but for the sake of it lets workout some values for what e can be

multiples of 9 are 9,18,27, 36, 54 63, 72, 81, 90 etc

if e= 5 , then d is a multiple of 9 because 5*5 - 7 = 18 which is a multiple of 9

if e = 14 then 5*14 = 70 and 70-7 = 63 which is a multiple of 9 therefore e can also be 14

S2 alone tells us that d= 5e+ 2

this is also clearly not sufficient on its own.

S1 + S2 plug in the values for e we found in S1 into S2 and see if the resulting value is a multiple of 9

5*5 + 2 = 27 --> multiple of 9
5*14 + 2 = 72 --> multiple of 9

We can then conclude the answer is C using both statements as whatever value we get for d is a multiple of 9

note that d can be 27 or 72 but either way it is a multiple of 9.

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