GMAT Prep - The positive integer k has exactly two...

[Mike]

The positive integer k has exactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, including 1 and k, what is the value of k?

1) 3^2 is a factor of k.
2) 7^2 is NOT a factor of k.

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GMAT Prep states that "Each statement Alone is sufficient".
Why the first is sufficient?

[DaveGill]

Use the rule: For a given number all the combinations of its prime factors are factors of that number.
K has 6 factors: 1, 3, 7, _, _, k (4 factors are known)
7x3 is also a factor by the above rule.
so the factors are : 1, 3, 7, 21, _, k


Statement 1
3^2 is factor, so the last unknown factor is 9.
so the factors are : 1, 3, 7, 21, 9, k
There are two 3s and one 7 as prime factors: so the number is 3x3x7 = 63
SUFFICIENT

Statement 2
7^2 is NOT a factor of k. (so only one 7 as a primefactor).
The missing factor will have to formed with other primefactor : so 3x3 is last missing factor.
There are two 3 and one 7 as prime factors: so the number is 3x3x7 = 63
SUFFICIENT

[RonPurewal]

Use the rule: For a given number all the combinations of its prime factors are factors of that number.
K has 6 factors: 1, 3, 7, _, _, k (4 factors are known)
7x3 is also a factor by the above rule.
so the factors are : 1, 3, 7, 21, _, k


Statement 1
3^2 is factor, so the last unknown factor is 9.
so the factors are : 1, 3, 7, 21, 9, k
There are two 3s and one 7 as prime factors: so the number is 3x3x7 = 63
SUFFICIENT

Statement 2
7^2 is NOT a factor of k. (so only one 7 as a primefactor).
The missing factor will have to formed with other primefactor : so 3x3 is last missing factor.
There are two 3 and one 7 as prime factors: so the number is 3x3x7 = 63
SUFFICIENT

nice explanation dave.

you can also figure this out in the following more generic way:
let's say you have 'M' number of 3's, and 'N' number of 7's.
this means there are M + 1 possibilities for the number of 3's in a factor of k (no 3's, one 3, ..., 'M' 3's), and, likewise, there are N + 1 possibilities for the number of 7's (no 7's, one 7, ..., 'N' 7's).
therefore, the total number of factors is (M + 1)(N + 1). (this formula generalizes to more than 2 prime factors as well, although it's unlikely the gmat would hit you with a problem involving such generalizations)
if there are 6 factors, this product is either 2x3 or 3x2, meaning that there are either one 3 and two 7's, or two 3's and one 7. (you can't have zero 3's and five 7's, because then 3 isn't a prime factor at all; likewise for five 3's and zero 7's.)
this means the only possibilities for k are 3x3x7 and 3x7x7, before you even examine statements (1) and (2). go from there.

i will readily admit this approach is a bit of a pain for numbers this small, but it's nice for larger numbers: you can instantly figure out, for instance, that the number (5^10)(3^8) has a total of 11 x 9 = 99 different factors. that could possibly come in handy.

[PudGe]

Ron,
Couldn't help but reply to this 4 yr old thread.
Thanks for your incredibly suave approaches to official questions-they are really useful and a great value addition to my prep.

thanks Manhattan team!

[tim]

:)