Divisor rules - If 15 and 4 are factors of positive integer

[Brent@GMATPrepNow]

If 15 and 4 are factors of positive integer K, and neither 5 nor 8 is a factor of positive integer M, then K – M CANNOT equal

A) 34
B) 44
C) 54
D) 70
E) 83

Answer: D

Source: www.gmatprepnow.com
Difficulty level: 600-650

[elias.latour.apex]

Since number K will be a multiple of 15 and 4, it must end in a zero. Numbers such as 60, 120, and 180, for example, fit the bill.

To reach answer choice (D), you would have to subtract a number ending in zero. That is to say that the number must be divisible by 10. All numbers divisible by 10 are also divisible by 5.

However, I must say that the question is not as well worded as it could be. One might conclude from the question that since 5 and 8 are not factors of 30, that 30 is okay because although it's divisible by 5, it isn't divisible by 8.

The question would have been clearer had the question said:

... neither 5 nor 8 is a factor of ...

[Brent@GMATPrepNow]

... neither 5 nor 8 is a factor of ...

Good suggestion. I've edited the question.

Cheers,
Brent

[M Matt@VeritasPrep]

K = 60*n, where n is some integer whose value we don't care about.

From here, let's try each answer:

(A) 60n - m = 34, how about n = 1 and m = 26?
(B) 60n - m = 44, how about n = 2 and m = 76?
(C) 60n - m = 54, how about n = 1 and m = 6?
(D) uh oh!
(E) 60n - m = 83, how about n = 2 and m = 37?

Using a few simple numbers, we find that (D) is the only answer without an easy solution, so it must be right.

[M Matt@VeritasPrep]

We could also be a little more rigorous and try to disprove (D) as a possibility altogether.

If 60n - m = 70, then we know m = 60n - 70 = 10 * (6n - 7).

Since m = 10 * something, m must be divisible by 10. But we're told that 5 is not a factor of m, so this is impossible, and we're set.

[Brent@GMATPrepNow]

If 15 and 4 are factors of positive integer K, and neither 5 nor 8 is a factor of positive integer M, then K – M CANNOT equal

A) 34
B) 44
C) 54
D) 70
E) 83

Answer: D

Source: www.gmatprepnow.com
Difficulty level: 600-650

RULE #1: If d is a divisor/factor of x, but d is NOT a divisor/factor of y, then d is NOT a divisor/factor of x+y
RULE #2: If d is a divisor/factor of x, but d is NOT a divisor/factor of y, then d is NOT a divisor/factor of x-y


If 15 is a factor of K, then we can also be certain that 5 is a factor of K.
So, we know that 5 is a factor of K AND we know that 5 is not a factor of M
By RULE #2, we know that 5 is not a factor of K - M
So, K-M cannot equal 70

Answer: D

[M Matt@VeritasPrep]

RULE #1: If d is a divisor/factor of x, but d is NOT a divisor/factor of y, then d is NOT a divisor/factor of x+y

Just to illustrate:

Suppose we take 3 and 5. If we think of 5 in terms of 3, we could say 5 = 3 + 2. So when we add them together, we get

3 + 5 =>

3 + (3 + 2)

and the resulting number WON'T be divisible by 3, because of that remainder of 2 that's hanging around.

Another example:

9 + 16 =>

9 + 15 + 1 =>

3*3 + 5*3 + 1

and again we've got that remainder.

[M Matt@VeritasPrep]

If we want to illustrate Brent's point algebraically:

x = d * n, where n is some integer we don't care about

y = d * m + r, where m is some integer we don't care about, and r is the nonzero remainder when y is divided by d

x + y =>

d*n + d*m + r =>

d*(n + m) + r

So we've got a multiple of d ... plus a remainder, which makes it no longer a multiple of d.

[Scott@TargetTestPrep]

If 15 and 4 are factors of positive integer K, and neither 5 nor 8 is a factor of positive integer M, then K – M CANNOT equal

A) 34
B) 44
C) 54
D) 70
E) 83

Answer: D

Source: www.gmatprepnow.com
Difficulty level: 600-650

Since 15 (and hence 5) is a factor of K, if 5 is also a factor of M, then 5 would be a factor of K - M. However, since 5 is not a factor of M, K - M can’t be divisible by 5. Therefore, K - M can’t be 70.

Answer: D