If a and b are positive integers, is a a multiple of b?

[puneet]

DS Question:

If a and b are positive integers, is a a multiple of b?

1. Every prime factor of b is also a prime factor of a
2. Every factor of b is also a factor of a

Source: Princeton - Cracking the GMAT - Practise Test

Can someone answer this question and explain it?

[AG]

Rephrase the Q. is b a factor of a?

from (1) lets say a = 12 and b=18

prime factors of a = 2 and 3
prime factors of b = 2 and 3

so b is not a factor

if we take b = 6 and a = 12

b is a factor so insufficient

from (2) using the same logic

b=18 gives factors = 2,3,3
a = 2 x 3 x 3 x ? which means b is a factor of a irrespective of what b is so sufficient

[RonPurewal]

DS Question:

If a and b are positive integers, is a a multiple of b?

1. Every prime factor of b is also a prime factor of a
2. Every factor of b is also a factor of a

Source: Princeton - Cracking the GMAT - Practise Test

Can someone answer this question and explain it?

the above poster pretty much has the essence of this one, but here are a few additional comments.

(1)
all you have here is information about prime factors; that means you have no idea HOW MANY of those prime factors there are. the trouble here, then, is that you could easily switch the roles of a and b: this statement is true if a = 4 and b = 8 (because the only prime factor is 2), but it's also true if you switch them to a = 8 and b = 4. so that's insufficient.

(2)
here's the easiest way to figure this one:
b is a factor of itself. therefore, it has to be a factor of a.
done. isn't that awesome?
if you don't think of that, you can always try plugging different numbers; after you discover enough times in a row that b has to go into a, you should start believing that it has to happen in general.

[puneet]

Thanks Ron.

What got me off track was that I WAS considering the number of times a factor occurs in my evaluation (of "Every prime factor of b is also a prime factor of a") and hence options 1 and 2 appeared to be similar. Your answer clarifies this.

[RonPurewal]

Thanks Ron.

What got me off track was that I WAS considering the number of times a factor occurs in my evaluation (of "Every prime factor of b is also a prime factor of a") and hence options 1 and 2 appeared to be similar. Your answer clarifies this.

understandable - but, unfortunately, still incorrect. whenever you go over a problem like this, make sure you hold on to any new insights about the meaning / usage of terms.

good luck

[saurabhbhardwaj]

Thanks Ron,

But I still have a 2 question:

(1)
all you have here is information about prime factors; that means you have no idea HOW MANY of those prime factors there are. the trouble here, then, is that you could easily switch the roles of a and b: this statement is true if a = 4 and b = 8 (because the only prime factor is 2), but it's also true if you switch them to a = 8 and b = 4. so that's insufficient.

The question asks is a a multiple of b?

Q1. In the above poster how are you saying "this statement is true if a = 4 and b = 8"? as I understand a <4> is not a multiple of b <8>

Q2. What are the prime factors of 12? As I understand, Prime factors of 12 are 2, 2, 3. So if 12 is "a" as in the above question and and 6 is "b" as in the above question, I think it will always be true that all prime factors of "b" (2,3) will be prime factors of "a" and hence a will always be a multiple of b?

Please correct if my understanding is incorrect.

[cyber_office]

Hi,

I suspect that he meant it is false in the first case. Both 4 and 8 share the prime factor of 2. Therefore,

A = 4 and B = 8; False

A = 8 and B = 4; True

Hence, indeterminate. Insufficient.

[RonPurewal]

Q1. In the above poster how are you saying "this statement is true if a = 4 and b = 8"? as I understand a <4> is not a multiple of b <8>

statement (1) is true for those values. since we're considering the sufficiency of statement (1), we need to select only values that satisfy that particular statement.

a = 4, b = 8, satisfy statement (1), but give a NO to the prompt question.
a = 8, b = 4, also satisfy statement (1), but give a YES to the prompt question.
therefore, statement (1) is insufficient.

don't confuse "statement" with "prompt / question". we will NEVER use "statement" to refer to the question.

Q2. What are the prime factors of 12? As I understand, Prime factors of 12 are 2, 2, 3. So if 12 is "a" as in the above question and and 6 is "b" as in the above question, I think it will always be true that all prime factors of "b" (2,3) will be prime factors of "a" and hence a will always be a multiple of b?

i don't really see what you're trying to say here. however, your example is a = 12, b = 6, which is functionally identical to my example of a = 8, b = 4.

...but i'm a bit troubled by this:
and hence a will always be a multiple of b?
so wait - do you think that one example proves that something is ALWAYS true? if so, that's going to be a big problem.