IF the integer n is greater than 1, is n equal to 2?

[guest]

Interesting DS Question:

IF the integer n is greater than 1, is n equal to 2?

(1) n has exactly two postive factors

(2) The difference between any two distinct positive factors is odd.

Answer: B

It looks easy but I did not get it right the first time. Here is my approach:

(1) Insufficient: N has exactly two positive factors, hence any prime would fit the constraint
(2) Sufficient: Any number would have 1 and itself as factors. Hence, N has to be even in order for N-1 to be odd. Examples:
2 has two factors (1 and 2). Rule holds. 4 has three factors (1, 2 and 4). Rule doesn't hold. Same is true for any even number greater than two. Hence, B is sufficient.

[Guest]

Statement 2:
In case of the number 4, the factors are 1,2 and 4.
4-1 = 3 (odd)
Similarly for factors of 12, 6-1 = 5 (odd).
Hence insuffcient.

Both together mean the number is 2 and sufficient. Should'nt the answer be C ?

[RonPurewal]

this is a really cool question.

(1)
this is a disguised way of saying 'n is prime'
therefore, insufficient

(2)
this says any two factors. that means any two factors - i.e., ALL pairs of factors have an odd difference.
there's only one way to do this: one odd factor and one even factor. (as soon as you get 2 odd factors or 2 even factors, you get an even difference by subtracting them.)
2 is the only # with only 1 odd factor and only 1 even factor.
therefore, sufficient

[Guest]

I don't think that is completely true... Can you explain 6?

If I factor 6 I can get

3-2 = 1 (odd)
or 6 - 1 = 5 (odd)

I don't understand why 2 is the only answer suff for B.

[tmmyc]

You need to look at all possible pairs:

3-2 = 1 (odd)
6-1 = 5 (odd)
2-1 = 1 (odd)
3-1 = 2 (even)
6-2 = 4 (even)
6-3 = 3 (odd)

This can be done for any integer. You can see that all numbers greater than 2 will have at least 1 odd and 1 even difference.

[RonPurewal]

You need to look at all possible pairs:

3-2 = 1 (odd)
6-1 = 5 (odd)
2-1 = 1 (odd)
3-1 = 2 (even)
6-2 = 4 (even)
6-3 = 3 (odd)

This can be done for any integer. You can see that all numbers greater than 2 will have at least 1 odd and 1 even difference.

correct.

remember to be very literalwhen you read and interpret mathematical terms. if a problem says 'pair of factors, that's ALL it means: two factors. in particular, it does NOT mean 'two factors that multiply to give the original number', which is the assumption that appears to motivate the previous post.

[badrinarayanang]

We now understand the OG's explanation, thanks to this forum. However, "any" could be interpreted as "atleast one pair", (not necessarily the pairs of factors that multiply with each other to give the original number), whose difference is odd. So when we found 1 as the difference of (3,2), the factors of 12, we felt here is a case which obeys statement 2. So we went on to infer that this statement is possible only when the factors are 1 and the number itself, implying C as the correct answer.

The point is that terms like "any" may be interpreted in more than one way. Had the question been something like "every" or "for all", we could have understood it far better. Now what's the solution? Assuming "any" to mean "every" whenever we come across "any" again in math?

Any comments on possibilities of such ambiguity in GMAT?

[mschwrtz]

Good question about ambiguity badrinarayanang, and the answer is that you should not understand "any" to mean " at least one" on the GMAT quant.

Similarly, you should not understand "Can n be expressed as, e.g. the product of two primes?" to mean "Is there an n such that n can be expressed as, e.g. the product of two primes?"

But whatever ambiguity you find in such expressions is not special to the GMAT.

[700+]

remember to be very literalwhen you read and interpret mathematical terms. if a problem says 'pair of factors, that's ALL it means: two factors. in particular, it does NOT mean 'two factors that multiply to give the original number', which is the assumption that appears to motivate the previous post.

Statement 2 says The difference between any two distinct positive factors is odd

I'm still very much confused about the 2nd statement. Could you please explain it again.

[RonPurewal]

remember to be very literalwhen you read and interpret mathematical terms. if a problem says 'pair of factors, that's ALL it means: two factors. in particular, it does NOT mean 'two factors that multiply to give the original number', which is the assumption that appears to motivate the previous post.

Statement 2 says The difference between any two distinct positive factors is odd

I'm still very much confused about the 2nd statement. Could you please explain it again.

hi -- please tell me what you didn't understand about the explanation (i assume you're talking about the one given here: post7645.html#p7645). without such specifics, i would probably just wind up giving the same explanation again.

thanks.

[michail.palagaschwili]

This one was very tough, but I got it right and could observe interesting number properties regarding this question.

(1) It says that n is a prime number. Clearly not sufficient
(2) Every odd > 1 number has ONLY odd factors, so the difference between factors will be always EVEN (ODD-ODD=EVEN)
Every even numer > 2 has both ODD and EVEN (# of even factors ≥ 2) factors, so their difference can be ODD and/or EVEN
And ONLY the difference of any two distinct positive factors of 2 is ODD

Math experts please correct me if I'm wrong.

[RonPurewal]

that all seems accurate.

it's hard to understand what you wrote -- especially because there are so many words in random boldface/all caps, for no apparent reason. but, from what i can tell at least, it all looks correct.

[RonPurewal]

by the way—
the most important thing is what you wrote in the first line -- "observe interesting number properties". especially that first word, OBSERVE.
it makes me almost deliriously happy to see that word there, because that's pretty much the whole point of these problems!
i.e., the point is for you to see patterns in the examples that you consider.

way, way, way too many people think that they have to KNOW or MEMORIZE these properties.
first, that's unrealistic (it's impossible to memorize every meaningful property of numbers that could be tested). second, it's actually counterproductive (it's harder to observe patterns if your head is full of spurious "rules").
so, it's excellent to see someone taking a better approach!

[michail.palagaschwili]

Hi Ron,

thanks, I was glad to see your comment. I hope that I'll be able to see these patterns in the real exam on January, 25th.
I've invested really much time in the preparation, but have still immense problems with RC, so everything above 37 in the verbal part would make me happy)).

[RonPurewal]

good luck.