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

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If p is the product of the integers from 1 to 30, inclusive [#permalink] New post 20 Jun 2010, 22:59
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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
[Reveal] Spoiler: OA
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Re: OG12 - PS [#permalink] New post 21 Jun 2010, 01:23
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divakarbio7 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

Not sure as to how to approach this problem


Given: p=30!=3^k*n. Question: k_{max}=?.

We should find highest power of 3 in 30!. Finding the power of a prime in n!: everything-about-factorials-on-the-gmat-85592.html or math-number-theory-88376.html (factorials chapter).

So:

\frac{30}{3}+\frac{30}{9}+\frac{30}{27}=10+3+1=14.

Answer: C.
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Re: official guide 13th ed #116 problem solving [#permalink] New post 19 Jun 2012, 05:13
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peter2000 wrote:
p is product of integers from 1 to 30 inclusive, what is the greatest integer k for which3^k is a factor of p?

A)10
B)12
C)14
D)16
E)18

I understand the way the GMAT guide does the problem, I was wondering if there was a quicker way?


Check out this post for an explanation of the method you can use to solve such questions:
http://www.veritasprep.com/blog/2011/06 ... actorials/
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new [#permalink] New post 23 Jun 2010, 12:38
15. No need to actually find the product of 1 to 30. Just look at the numbers that have factors of 3.

3 = 3
6 = 3 * 2
9 = 3 * 3
12 = 3 * 2 * 2
15 = 3 * 5
18 = 3 * 3 * 2
21 = 3 * 7
24 = 3 * 2 * 2 * 2
27 = 3 * 3 * 3
30 = 3 * 2 * 5

There are 14 3's, so the largest factor of p for 3k is 3*14.
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new [#permalink] New post 26 Feb 2012, 10:19
\frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}
= 10 + 3 + 1 (ignore the decimals part of each fraction)
= 14
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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink] New post 01 Oct 2012, 05:50
P = 30!
8
P = 30 x 29 x 28 x 27 x 26 x 25 x 24 x 24 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 09 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1


Out of these 30 , 27 , 24 , 21 , 18, 15 , 12 , 09 , 06 , 3 are factors of 3

3 x 10 , 3 x 3 x 3 , 3 x 8 , 3 x 3 x 2, 3 x 5 , 3 x 4 , 3 x 3 x 3 , 3 x 2 , 3

So we have a total of 14 three's ...

Therefore the maximum value of K can be 14 (C)
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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink] New post 23 Oct 2013, 23:58
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Re: If p is the product of the integers from 1 to 30, inclusive   [#permalink] 23 Oct 2013, 23:58
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