Problem Solving

Thanks for your support in this one..


If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

Hi,

Find all multiples of 3 upto 30.
P = (3,6,9,12,15....30)
There are 10 of them.
P = 3^10(1,2,3,4...10)

Now look for multiples of 3 again.. 3,6,9.
There are 4 3's.

So 3 ^14 is a factor of p. IMO k=14

imane81 wrote:Thanks for your support in this one..


If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

Hi,

we're asked how many 3s are factors of 30!.

We start by noting that 30/3 = 10, so there are 10 multiples of 3 in the factorial. 10 is meant to be a very tempting answer.

However, we should also note, than 9, 18 and 27 are multiples of 9, which has a second 3 in its factorization; that adds 3 more 3s to our total.

Finally, 27 has a 3rd 3 in its factorization, bringing our grand total to 14 3s: choose C.