Factors anD factors

[ve_boss]

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

[Dynamo]

IMO D

K has 6 factors, 3,7,k,1, and 2 others either 3 or 7

statement 1

3^2 is a factor of K therefore K = 3^2*7^2 = 63

Similarly statement 2

7^2 is not a factor of K therefore K = 3^3*7 = 159

[ve_boss]

@Dynamo,

I'd like to have your opinion on (1).
When it's said :3^2 is a factor of k, Does that necessarily means that 3^3 is not a factor of k ?

In fact, the factors of k could be : 1,3,3,3,7,k
In this case 3^2 is effectively a factor of k but k is not 63.

Thanks

[Dynamo]

Hmmm good observation I missed that out... I assumed in a very silly way...

so in that case it should be C...

[ve_boss]

My answer was B, but the Official Answer (in GmatPrep) is D. It's confusing...

[krovvidy]

k can be represented as (3^m)*(7^n) where m & n are integers >=0

Total no. of factors = (m+1)*(n+1) = 6

(1) Says 3^2 is a factor i.e., m>=2 ----> SUFFICIENT because (m,n) can only be (2,1) to satisfy (m+1)*(n+1) = 6

(2) Says 7^2 is not a factor i.e., n<2 ----> SUFFICIENT because (m,n) can only be (2,1) to satisfy (m+1)*(n+1) = 6


Hence D is correct ...

[bose]

amazing krovvidy, didn't know this formula. thanks.imo d. i get it 3^m will have factors 1,3,3^2...,3^m so in all (m+1) similarly 7^n will have factors (n+1) so each one of the factors in the fisrt set can be multiplied by each factor in the second set to form a factor, so we have total (m+1)(n+1) factors.