Manhattan GMAT 5thED Number Properties

[sid.mohan84]

I am unable to understand the approach/explanation provided for Chapter 1, problem #4.

The only difference I can make between #1 and #4 are the words "or" and "and" respectively. Can somebody care to explain.

I am stating both problems below as well.

#1. If a is divided by 7 or 18, an integer results. Is a/42 an integer?
#4. If j is divisible by 12 and 10, is j divisible by 24?

Thank you.

[[email protected]]

Hi sid.mohan84,

These two sentences mean the exact same thing.

#1 If a is divided by 7 OR 18, an integer results.

This means that a is evenly divisible by both 7 and 18. It has to be, since an integer results in either situation.

#4 If j is divisible by 12 AND 10....

This means that j is evenly divisible by both 12 and 10.

GMAT assassins aren't born, they're made,
Rich

[sid.mohan84]

Thank you Rich for responding. However, the solutions section provides two different answers for the problems.

For #1, the answer is YES.
For #4, the answer is CAN NOT BE DETERMINED.

Can you please explain, why? if I use teh prime factor technique then it must be YES for both problems.

-Sid

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Hi Sid,

I was under the impression that you were asking about the words "or" and "and" in context of the two phrases. The questions themselves are both YES/NO questions and will be based on what you pick for "a" (in #1) and "j" (in #4).

As far as the math is concerned....

In #1, "a" must be a multiple of 7 and 18. You must determine which numbers are multiples of both. Here's a partial list:

126, 256, 378, etc.

42 is evenly divisible into all of these examples, so the answer to the question is ALWAYS YES. Consistent = SUFFICIENT

In #4, "j" must be a multiple of 12 and 10. Here's a partial list:

60, 120, 180, 240, etc.

24 does NOT evenly go into 60, but it DOES evenly go into 120. Sometimes NO, Sometimes YES = Inconsistent = INSUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

[sid.mohan84]

Thank you.

[sid.mohan84]

Rich. However, finding multiples of both numbers take quite some time. Is there a quick approach. The Manhattan book tells me that for

#1. If a number is divisible by 7 and 18, then the prime factors of that number are 2,3,3, and 7.
And since 42 can be constructed as a product of these numbers. The number is divisible by 42.

#2. here if I follow the same approach, the prime factors of 12 and 10 are 2, 2, 3, 2 and 5. And I can make 24 out of these numbers. Then why is the answer different in this case.

[[email protected]]

Hi Sid,

That "math approach" you're referring to is known as prime factorization, which MIGHT use one time on Test Day (it's not all that frequent or useful of an approach). Academically, you're multiplying numbers together; if that takes a long time to do, then you need to practice more (these basic skills require precision and speed; it's difficult to score 700+ unless you're perfect in the basics).

For #1, your factors are unique (there are no "shared" factors), so 2 x 3 x 3 x 7 = 84 (since 42 goes into 84, 42 will go into any multiple of 84) = CONSISTENT = SUFFICIENT

For #4 however, both 10 and 12 contain a 2, so you DON'T count it twice. You'd have 2 x 3 x 3 x 5 = 60 (24 doesn't go into 60, but it goes into 120) = INCONSISTENT = INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich

[sid.mohan84]

Clear as crystal. Thanks.

[Matt@VeritasPrep]

Two quick postscripts to this:

i) 2 * 3 * 3 * 7 isn't 84, it's 126, so the first statement tells you that the number is a multiple of 126. Since 126 is divisible by 42, the number itself must be divisible by 42. As an interesting side consequence (that the GMAT often tests), the number doesn't HAVE TO be divisible by 84, as to be divisible by 84 the number must have two 2's in its prime factorization, but it COULD be - we would need to know more about its other factors (as Rich explained when answering the second problem).

ii) Prime factorization is quite useful on the GMAT, IMHO: many number properties questions draw on it, particularly challenging DS or "how many of these three statements must be true" questions. I wouldn't say it's a tool you'll only use once, at least not if you're scoring well and approaching questions technically (rather than by picking numbers and hoping for a pattern). That two equal integers (greater than 1) share the same prime factorization is a property that comes up surprisingly often.