moneyman wrote:if the integers a and n are greater than iand 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
Ans B
i shall assume that the text is supposed to read as follows:
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?
--
as with just about all other divisibility problems, you need to break down the number in question - in this case the product of the first eight positive integers - into primes.
this is 2 x 3 x 4 x 5 x 6 x 7 x 8
= 2 x 3 x 2 x 2 x 5 x 2 x 3 x 7 x 2 x 2 x 2
= 2^7 x 3^2 x 5 x 7
--
statement (1)
this statement establishes a^n = 64, but the problem is that there are multiple pairs (a, n) that satisfy this criterion: 2^6, 4^3, 8^2. this statement is therefore insufficient.
interestingly enough, this statement renders the whole bit about 8! completely irrelevant, because it just hands you the value of a^n. (that statement is only useful if it helps you figure out what a^n is, as in the next part)
--
statement (2)
this statement says that some number
to the sixth power is a factor of 8!. this means that some number
must appear at least six times in the prime factorization above. the only number 'a' that comes anywhere close to satisfying this condition is a = 2.
sufficient.
--
answer = b
--
also, note the following general piece of wisdom:
on a problem such as this one, on which the two statements together are OBVIOUSLY sufficient, you should be extremely suspicious of choice c.
remember, these guys are tricky. in this problem, it is crystal clear that the two statements together are good enough (they give a^6 = 64, so a is definitely 2);
they would very rarely write a problem with an answer that obvious, so, if you have to guess on a problem like this, DON'T guess c (or e, which is impossible). instead, bank on at least one of the 2 statements being sufficient, and so guess a, b, or d.
