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# If p is the product of the integers from 1 to 30, inclusive, what is t

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If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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09 Aug 2009, 12:10
1
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14
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69% (01:51) correct 31% (01:16) wrong based on 398 sessions

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Sep 2014, 07:30, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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09 Aug 2009, 14:02
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??
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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09 Aug 2009, 23:04
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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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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09 Aug 2009, 23:22
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: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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10 Aug 2009, 14:45
No....OA is C

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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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11 Aug 2009, 04:04
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: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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11 Aug 2009, 04:32
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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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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30 Aug 2009, 00:18
1
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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, 15:29, edited 1 time in total.
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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30 Aug 2009, 06:52
1
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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: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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29 Sep 2014, 07:27
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

Can we change 3k to 3^K . A minor correction to make this question correct.

Tx
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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29 Sep 2014, 07:30
5
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Expert's post
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tarek99 wrote:
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

Finding the number of powers of a prime number k, in the n!.

The formula is:
$$\frac{n}{k}+\frac{n}{k^2}+\frac{n}{k^3}$$ ... till $$n>k^x$$

For example: what is the power of 2 in 25! (the highest value of m for which 2^m is a factor of 25!)
$$\frac{25}{2}+\frac{25}{4}+\frac{25}{8}+\frac{25}{16}=12+6+3+1=22$$. So the highest power of 2 in 25! is 22: $$2^{22}*k=25!$$, where k is the product of other multiple of 25!.

Check for more: everything-about-factorials-on-the-gmat-85592.html and math-number-theory-88376.html

Back to the original question:
If p is the product of 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

Given $$p=30!$$.

Now, we should check the highest power of 3 in 30!: $$\frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}=10+3+1=14$$. So the highest power of 3 in 30! is 1.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]

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03 Aug 2015, 19:52
1
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Thanks
tarek99 wrote:
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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html

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Re: If p is the product of the integers from 1 to 30, inclusive, what is t   [#permalink] 03 Aug 2015, 19:52
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