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Divisibility and Primes (MGMAT Practice Set Problem)

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Divisibility and Primes (MGMAT Practice Set Problem)

by ostrowskiamy » Tue Feb 12, 2013 7:59 pm
Hi folks,

Hope all is well in study-land! I THOUGHT I had this concept nailed, but I incorrectly answered a practice problem in my MGMAT book, and I'm still stuck on how to determine if an answer is flat-out "no" versus "cannot be determined."

Question: (Original prompt: "Use one or more prime boxes, if appropriate, to answer each question: YES, NO, or CANNOT BE DETERMINED. If your answer is CANNOT BE DETERMINED, use two numerical examples to show how the problem could go either way. All variables are integers unless otherwise stated")

"If 80 is a factor of r, is 15 a factor of r"?

So, what I did was make a prime factor tree for 80, and ultimately ended up with: 2,2,2,2, and 5 as the prime factors of 80. Since no combination of those numbers creates a product of 80, I answered "no" to this question. However, the answer key reads: "CANNOT BE DETERMINED. If r is divisible by 80, it's prime factors include 2,2,2,2, and 5...15=3*5. Since the prime factor 3 is not in the box, we cannot determine whether 15 is a factor of r. We could take r=80, in which case 15 is NOT a factor, or r=240, in which case 15 IS a factor."

I understand, with the numerical examples, why it's a factor sometimes, but not all the time. BUT, I also thought that "my method" (creating a prime factor tree, determining the prime factors, and seeing if I could create the prime factor in question) sufficed, for determining if the answer is "yes" or "no." How can I quickly and efficiently distinguish (during the actual GMAT) if (in this case) 15 is flat-out "no" or just "possibly", since I won't have time to test all sorts of different multiples?

Thank you!

Amy
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by ostrowskiamy » Tue Feb 12, 2013 8:09 pm
Quick follow up to this: I incorrectly answered another questions as "yes" when it turns out the answer is "cannot be determined", so I would love some help on the "yes" part, too! For example:

"If j is divisible by 12 and 10, is j divisible by 24?"

My answer = YES. 12 = 2*2*3 and 10 = 5*2. 2*2*2*3 = 24.
Book's answer = CANNOT BE DETERMINED. "If j is divisible by 12 and by 10, its prime factors include 2, 2, 3, and 5, as indicated by the prime box to the left. There are only TWO 2's that are definitely in the prime factorization of j, because the 2 in the prime factorization of 10 MAY be redundant - that is, it may be the SAME 2 as one of the 2's in the prime factorization of 12." Since 24 requires three 2's, 24 is not necessarily a factor of j."

In theory, I get it...but, I can clearly see from the two prime factor trees I made that there are three 2's. When should I account for potential redundancy - in EVERY problem such as the one above, where I'm looking at the prime factorization of two different numbers and trying to determine if a third is also a factor of the number that the previous two are? I'm confident I can get this, but any nudge in the right direction for how to (always) correctly identify if redundancy should be concerned (and therefore, the number of prime factors I'm left with reduces) would be greatly appreciated! Thank you so much!

Best,
Amy
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by Ian Stewart » Wed Feb 13, 2013 4:28 am
I'll use simpler numbers for illustration, and then perhaps you can return to the questions above and see if they make more sense.

If you're told that x is divisible by 6, so x is divisible by 2*3, then that means 2 and 3 are some of the primes 'inside' of x. But there may be others; x might be equal to 6, but x might also be equal to 24, or to 6,000,000. So if a Data Sufficiency question asks:

Is the positive integer x divisible by 17?
1.x is divisible by 6
2.x is divisible by 34


then from Statement 1, we know that 2 and 3 are primes 'inside' of x, but they aren't necessarily the only primes 'inside' of x. Any other primes could be there as well, including 17 (x could be equal to 2*3*17 for example) so Statement 1 is not sufficient. From Statement 2, we know x is divisible by 2*17, so x is certainly divisible by 17, and the answer is B - Statement 2 alone is sufficient.

In your first question, I think you've assumed, upon reading that r is divisible by 80, that r was exactly equal to 80. If r is divisible by 80, we know that r is divisible by (2^4)(5), but r could be divisible by all kinds of other things too.

For your second question, if you ever know that some integer x is divisible by a list of numbers, that always means that x is divisible by the Least Common Multiple of that list. So if you know, say, that x is divisible by 4 and by 6, that means, mathematically, the same thing as: x is divisible by the LCM of 4 and 6. In other words, that means that x is divisible by 12. It's much easier to think about a single number than it is to think about a list of numbers, so by first finding the LCM, you will always find you have much simpler information to think about. So if you are asked the following:

Is the positive integer k divisible by 18?
1. k is divisible by 6 and by 9
2. k is divisible by 6 and by 15


in each statement, we are told that k is divisible by a list of numbers. It will be easier to think about each statement if we find the LCM of each list. So Statement 1 just tells us that k is divisible by the LCM of 6 and 9, which is 18, and that's precisely what we wanted to know. So Statement 1 is sufficient. Statement 2 tells us that k is divisible by the LCM of 6 and 15, which is 30. So we know k is divisible by 2*3*5, but we cannot be sure whether k is divisible by 18 = 2*(3^2), since we cannot say whether 3^2 is a factor of k (it may be, or it may not be), so the answer may be yes or may be no, and Statement 2 is not sufficient. So the answer would be A.

If in your second question, you first find the LCM of 10 and 12, you'll find that j must be divisible by 60. In that case, j might not be divisible by 24 (j might be exactly equal to 60, for example), but j might be divisible by 24 (j could be 120 or 240, say).
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by ceilidh.erickson » Wed Feb 13, 2013 9:37 am
One thing that you want to keep in mind with the idea of prime boxes is that, as Ian said, they tell you some of what a number contains, but not everything that number contains. If you knew that x was equal to 80, you'd know that it has only [2, 2, 2, 2, 5] in its prime box. But if you're told that x is divisible by 80 (or that 80 is a factor of x, x is a multiple of 80, 80 goes evenly into x, etc etc), you know that x contains all of the factors of 80... but it might also contain other stuff!

That's why I think it's important to keep a question mark in any prime box of a variable to remind you that there might be other stuff you don't know about.

Image

Could that question mark represent a 3? Maybe! If so, it would be divisible by 15. If not, it wouldn't be.
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by ceilidh.erickson » Wed Feb 13, 2013 9:48 am
To your second question, Ian's right that we can think of the LCM. But we can also think of this conceptually...

The GMAT is definitely trying to trick us here! If we're told that j is divisible by 2*2*3 and 2*5, why can't we assume that it's divisible by 2*2*2*3?? That would be reasonable, right? The problem, though, is that we don't know if the 2 from 10 is a new 2, or if it's the same 2 that we had from 12. Think of this potential prime box:
Image


This can be a tricky concept in math, but I bet it's something you intuit in real-life scenarios.

If one friend told you "I just saw two elephants walking down the street!", and then another friend told you "I just saw three elephants walking down the street!" Assuming that your friends aren't insane or lying to you, would you conclude that there are 5 elephants walking down the street? No, you'd think "there must be at least 3 elephants, but maybe there are more."

The same idea applies here. We know that there are at least two 2's, one 3, and one 5, but we don't know if we have three 2's.
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Harvard Graduate School of Education
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