if the integers a and n are greater than 1 and the product of the first 8 positive integers is a multiple of a^n, what is the value of a?

(1) a^n = 64

(2) n = 6

Can someone break this down in a methodical manner? Thanks!

I ran into the same question.... Can anyone provide their soln?

Is the answer B?

If no, don't read the rest of this response.

The prime factorization for the product of the first 8 positive integers is (2^7)*(3^2)*5*7. The first statement alone is not sufficient because "a" can have a value of 2, 4 or 8 given that a^n=64. The second statement alone, however, is sufficient because given that n=6, the only integer value "a" for which a^n will still be a factor of (2^7)*(3^2)*5*7 is 2.

Let f(n) = Product of first 8 = 8 *7 *6*5*4*3*2 *1
That means a^n is a factor of f(n)

Express f(n) in form of prime factors
f(n) = (2)^3 * 7 * (2)^1 *(3)^1 * 5 * (2)^2 *3 * 2 * 1

= (2)^6 * 7 * 5 * (3)^2



(2)Take statement (2) first as it is easier

if n= 6, then only value which should satisy for a would 2 and 1, however a >1 and hence a =2
hence (2) is sufficient

(1) a^n = 64
This is possible only when a= 2 n= 6.
So this is also sufficient.

Hence ,the answere should be (D).

Please let me know if that's correct.

Hey guys -

Ran into this question a while back as well and below is my approach, hope it's helpful....

In analyzing the question stem, you can see that if 8! is a multiple of a^n, the prime factorization will, at the very least, have to contain a^n. If you break down the prime factorization of 8!, you get 2^3 x 7 x 3 x 2 x 5 x 2^2 x 3 x 2 (8 x 7 x 6, etc....), or simplified 2^7 x 3^2 x 5 x 7. We can now look at the statements to see which one might work....

Starting with statement 2, we see the exponent of a^n equals 6. Looking back at our simplified prime factorization, we can see that only 1 term is raised to the power of 6 or greater (2^7). Therefore, we can definitively conclude that a MUST equal 2, since we are told a^n is a factor of 8! (note that A could also have been equal to 1, had it not been for the constraint in the question stem). So we have it down to B and D.

I found statement 1 to be bit trickier. We are told a^n equals 64. Since we've been dealing with 2's already, it may be easier to see a^n as 2^6. However, 64 can also be expressed as 4^3. We have two possible values for a, thus yielding it insufficient.

Believe the answer was indeed B, but did this a while back so please correct me if I'm doling out incorrect advice!!!

you are doling out fantabulastic, wonderfulicious advice, actually. the content is largely the same as that of the posts above yours, but your style is lucid and easily understandable. well done.

one thing i should throw in on this problem: this is a classic case of the so-called "C TRAP".

here's what the 'c trap' is: if you look at both statements together, then they're obviously sufficient. in this case, if you take both statements together, you don't even need the question stem (!), apart from the part restricting 'a' to positive numbers - you wind up with a^6 = 64, so 'a' must be 2.

whenever you see a 'c trap' question, you can pretty much rest assured that the answer is NOT 'c'. so if you're forced to guess on a problem like this one, think about whether the two statements together form an obviously complete and sufficient pair; if they do, then, more than likely, one of them alone will be sufficient (and thus your guessing will be limited to (a), (b), and (e) ).

for other examples of 'c traps', open up your og11 to the data sufficiency section and check out problems #124, #129, and #150. (as usual, please do not comment on those problems, or transcribe any parts of them, in this thread. thank you)