sukhman wrote:Find the greatest number , which will divide 215, 167 and 135 so as to leave the same remainder in each case A. 64 B. 32 C.24 D.16

Why are you using the spoiler function to hide the answer choices? On the GMAT, the answer choices are always in plain sight.
Instead, we ask that you using the spoiler function to hide the correct answer so that others can attempt the question without seeing the final answer.

Cheers,
Brent

Find the greatest number , which will divide 215, 167 and 135 so as to leave the same remainder in each case
A. 64
B. 32
C. 24
D. 16

Notice that the remainder we get when we divide N by d is the same as when we divide N+d by d.
For example, when we divide 16 by 7, we get remainder 2. When we divide (16+7) by 7, we get remainder 2.
Likewise, when we divide 53 by 10, we get remainder 3. When we divide (53+10) by 10, we get remainder 3.

We can extend this property to say that the remainder we get when we divide N by d is the same as when we divide N+kd by d (where k is some integer) In other words, N and N+kd will have the same remainder when we divide both by d

We can take all of this information and conclude that, If A > B, and A and B have the same remainder when divided by d, then A - B = some multiple of d. In other words, d is a divisor of (A-B)

Okay, now onto the question. . .

Our three numbers are 135, 167, and 215

Notice that 167 is 32 greater than 135 (167 - 135 = 32).
So, in order for 135 and 167 to have the same remainder, we must divide both by a number that is a divisor of 32
Since 64 and 24 are NOT divisors of 32, we can eliminate A and C

Also notice that 215 is 48 greater than 167 (215 - 167 = 48)
So, in order for 167 and 215 to have the same remainder, we must divide both by a number that is a divisor of 48
Since 32 is NOT a divisor of 48, we can eliminate B

This leaves us with D, the correct answer.

Cheers,
Brent

sukhman wrote:Find the greatest number , which will divide 215, 167 and 135 so as to leave the same remainder in each case
A. 64
B. 32
C. 24
D. 16

ASIDE: I'd say that this question is flirting with the upper limit of GMAT quant questions. It's definitely 700+. Having said that, we can also test the answers by using some old fashion division.

Since we're looking for the GREATEST number, we'll start with the GREATEST answer choice.

A) 215 divided by 64 leaves remainder 23. 167 divided by 64 leaves remainder 39.
Not the same. ELIMINATE A

B) 215 divided by 32 leaves remainder 23. 167 divided by 32 leaves remainder 7.
Not the same. ELIMINATE B

C) 215 divided by 24 leaves remainder 23. 167 divided by 24 leaves remainder 23. 135 divided by 24 leaves remainder 15.
Not the same.

AT THIS POINT, I'd normally stop and assume that the answer must be D.

But let's check D for "fun"

D) 215 divided by 16 leaves remainder 7. 167 divided by 16 leaves remainder 7. 135 divided by 16 leaves remainder 7. IT WORKS!

The correct answer is D.

Cheers,
Brent