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# factor

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Joined: 21 Jul 2006
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factor [#permalink]  09 Aug 2009, 11:10
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Difficulty:

5% (low)

Question Stats:

80% (02:55) correct 20% (00:00) wrong based on 8 sessions
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

please show the fastest way to solve this.
thanks
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Kudos [?]: 27 [0], given: 6

Re: factor [#permalink]  09 Aug 2009, 13:02
I can see that any of these numbers is ok.
so I just pick 18.

P = 30! = 30x29x28x......x4x3x2x1

if 3K is a factor of p, it means that we can divide p by 3K and the result is an integer.
there is a "3" in the expression of p, and all numbers shown are also there.

the greatest one should then be OK.

is AO E??
Director
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Kudos [?]: 193 [0], given: 18

Re: factor [#permalink]  09 Aug 2009, 22:04
k should be the largest prime <=30. So k can be 29 !! as 3 * 29 is a factor of 30!.
However this option is not listed!!
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Kudos [?]: 52 [0], given: 12

Re: factor [#permalink]  09 Aug 2009, 22:22
Agree with Economist. But from the list, the best answer would be 18. Since all the prime factors of 18*3(2*3*3*3) would be in the product number. OA pls!
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Re: factor [#permalink]  10 Aug 2009, 13:45
No....OA is C

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Re: factor [#permalink]  11 Aug 2009, 03:04
1
KUDOS
tarek99 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

please show the fastest way to solve this.
thanks

Well ..I think question should read as 3^k ....i.e, 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?

Then C ..14 will be correct..
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Re: factor [#permalink]  11 Aug 2009, 03:32
1
KUDOS
age wrote:
tarek99 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18

please show the fastest way to solve this.
thanks

Well ..I think question should read as 3^k ....i.e, 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?

Then C ..14 will be correct..

I agree. If C is the OA, then the questions should read as 3^k giving 14 as the answer.

Although if its not 3^k, then another interpretation of 3k could be 300 + k instead of 3*k, because 3*k makes the question silly. This means that when k=10, we get 310 and not 3*10 = 30

Thus proceeding with the "300 + k" assumption, we get:
P = 30!

Option A: 310
=> 310 = 2 * 5 * 31
=> 31 is a prime and is not present in 30!
Thus 310 is not a factor of P
=> A is Eliminated

Option B: 312
=> 312 = 2^3 * 3 * 13
Thus 312 is a factor of P
=> B is the correct answer !

Option C: 314
=> 314 = 2 * 157
=> 157 is a prime and is not present in 30!
Thus 314 is not a factor of P
=> C is Eliminated

Option D: 316
=> 316 = 2^2 * 79
=> 79 is a prime and is not present in 30!
Thus 316 is not a factor of P
=> D is Eliminated

Option E: 318
=> 310 = 2 * 3 * 53
=> 53 is a prime and is not present in 30!
Thus 318 is not a factor of P
=> E is Eliminated

ANS: B
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Location: New York, NY
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Kudos [?]: 35 [0], given: 23

Re: factor [#permalink]  29 Aug 2009, 23:18
Factors of 31! include 14 times 3. Hence 3^14.
Number: Factors of 3
3 : 1
6 : 1
9 : 2
12 : 1
15 : 1
18 : 2
21 : 1
24 : 1
27 : 3
30 : 1
Total : 14

Last edited by mendelay on 30 Aug 2009, 14:29, edited 1 time in total.
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Kudos [?]: 89 [1] , given: 13

Re: factor [#permalink]  30 Aug 2009, 05:52
1
KUDOS
if it is 3*k, answer is E, just 18 in, 30! is divisible by 3*18

if it is 3^k, answer is C, 30!has 3, 6, 9,....30 as the factors, includes 14 of 3.
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Re: factor   [#permalink] 30 Aug 2009, 05:52
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