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New question - exponents

by Brian@VeritasPrep » Tue Nov 16, 2010 6:01 pm
Hey everyone,

I just drew up this question for a few of my students who wanted another exponent-based DS challenge and figured I'd share it here, too. Thoughts?

What is the value of product abc?

(1) 2^a * 3^b * 5^c=1728

(2) a, b, and c are nonnegative integers
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by nisha.menon294 » Tue Nov 16, 2010 8:12 pm
is it A ? - st 1 alone is sufficient

1) if you factorize 1728 -> 2^6 *3^3*5^0 . so the product is 0. Hence sufficient
2) Insufficient .
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by kuiper » Tue Nov 16, 2010 9:25 pm
I assume A as well

1728 is not multiple of 5 so c has to be 0 and hence the answer
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by Rahul@gurome » Tue Nov 16, 2010 9:37 pm
Brian@VeritasPrep wrote: What is the value of product abc?
  • (1) 2^a * 3^b * 5^c=1728
    (2) a, b, and c are non-negative integers
Statement 1: (2^a)*(3^b)*(5^c) = 1728
As nowhere it is mentioned that a, b, c are integers, there can be infinite number of solutions for a, b, c. For example take a = 1, b= 1. Then (5^c) = 1728/6 = 288 => c = log(288)/log(5). We can also take negative values for a, b, c and we will get valid results.

Not sufficient.

Statement 2: a, b, and c are non-negative integers.
There is infinite numbers of possible combination of a, b, c.

Not sufficient.

1 & 2 Together: (2^a)*(3^b)*(5^c) = 1728 and a, b, c cannot be fractions or negative.
Thus a, b, c must be the powers of 2, 3, 5 in the prime factorization of 1728 which is (2^6)*(3^3)*(5^0). Thus, a = 6, b = 3 and c = 0 => abc = 0.

Sufficient.

The correct answer is C.
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by fskilnik@GMATH » Wed Nov 17, 2010 1:36 am
Just a small alternative to guarantee that (1) is insufficient without going through "log" explicitly:

If you take (say) b = c = 0, from the fact that f(x) = 2^x is a function that has as its image the whole positive real numbers, there exists a such that 2^a = 1728... the product of abc would then be zero.

On the other hand, take (say) b=c=1 , again there exists a (not zero) such that 2^a = 1728/(3*5), and now abc is NOT zero.

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by Brian@VeritasPrep » Wed Nov 17, 2010 9:36 am
Hey guys,

Great discussion! And thanks, Fabio and Rahul for your contributions...you're right on. I like this question because it brings up a pretty big strategic point for DS questions which is:

Why Are You Here?

Regarding statement 2, it adds absolutely no value on its own...it's nowhere close to being sufficient on its own. So there are two likely reasons that it's there:

1) To trick you into thinking you need it, and therefore picking C instead of A
2) To add information that IS, in fact, necessary to go with statement 1, making the correct answer C and not A

The GMAT doesn't use "red herring" distraction statements very often at all - if they provide a statement in a DS problem there has to be a reason...it's either a trap or it's necessary. The good news for you is that you can use either case the same way - look at that statement to determine whether you really need it.

Here, although statement 1 may seem sufficient on its own (c must be 0 in order to invalidate the 5 term), that only fits if we know that they're all integers. Statement 2, by providing us with that information, should give us pause about statement 1: Do they have to be integers?

We don't need to use logarithms on the GMAT (thankfully!), but we should know enough that there would conceivably exist a set of noninteger exponents that would solve this problem. Like Fabio did, even if you just assume that a and b are 1 so that:

5^c = 288

There is some value for c that will get us 288, so we can prove that statement 1 is not on its own sufficient.


So, strategically, when you see a statement that on its own is clearly not sufficient, ask yourself "why are you here?". Is it providing essential information, or is it there to make you think you need it?
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by fskilnik@GMATH » Fri Nov 19, 2010 1:19 am
Very good points (and beautiful problem), Brian.

The conclusion, in particular,
Brian@VeritasPrep wrote:So, strategically, when you see a statement that on its own is clearly not sufficient, ask yourself "why are you here?". Is it providing essential information, or is it there to make you think you need it?
is important enough to be remember by all GMAT applicants in every DS problem.

Regards,
Fabio.

P.S.: for the very-technical student/reader, it is nice to observe that to guarantee the statements 1 and 2 are sufficient (together) to answer the question asked, we are in fact "going through" the Fundamental Theorem of Arithmetic: every integer greater than 1 may be factorized into primes in just one way if the order of primes is not taken into considerations. ;)
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