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Math Expert V
Joined: 02 Sep 2009
Posts: 58364
What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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Difficulty:   15% (low)

Question Stats: 75% (00:44) correct 25% (01:05) wrong based on 88 sessions

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What is the greatest integer p, for which 3^p is a factor of 21! ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

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Joined: 26 Mar 2013
Posts: 2345
Re: What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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Bunuel wrote:
What is the greatest integer p, for which 3^p is a factor of 21! ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

$$\frac{21}{3}$$+$$\frac{21}{9}$$=7 +2 =9

P = 9

Senior PS Moderator V
Joined: 26 Feb 2016
Posts: 3344
Location: India
GPA: 3.12
What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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Bunuel wrote:
What is the greatest integer p, for which 3^p is a factor of 21! ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

To get the greatest integer p, for which $$3^p$$ is a factor of 21!
we need to find how many 3's are there in 21!

3(1), 6(1), 9(2), 12(1), 15(1), 18(2), 21(1)

Total : 1+1+2+1+1+2+1 = 9
Therefore, the greatest value of p is 9(Option B)
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Re: What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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Bunuel wrote:
What is the greatest integer p, for which 3^p is a factor of 21! ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

To determine the number of factors of 3 within 21!, we can use the following shortcut in which we divide 21 by 3, and then divide the quotient of 21/3 by 3 and continue this process until we can no longer get a nonzero integer as the quotient.

21/3 = 7

7/3 = 2 (we can ignore the remainder)

Since 2/3 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 3 within 21!.

Thus, there are 7 + 2 = 9 factors of 3 within 21!.

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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
Re: What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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Solution:

Given:

• $$p$$ is an integer

• $$3^p$$ is a factor of $$21!$$

Working out:

We need to find the greatest value of “p

Since $$3^p$$is a factor of $$21!$$, and we need to find out the greatest integer p, it can be inferred that 3^(p+1) is not a factor of p.

• We need to calculate the number of 3s in 21!
Since all the numbers in 21! is in the multiplicative form, the powers of 3 will be added.

• Multiples of 3 in 21! are: 3, 6, 9, 12, 15, 18, 21.

o All the above 7 numbers have at least one 3 in them. However, 9 and 18 contain two 3s in them.

o Thus, the total number of 3s in 21! = 7+2 =9

Hence $$3^9$$ is a factor of $$21!$$ but $$3^{10}$$ is not.

Thus, the greatest integer value of p is 9

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Re: What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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To find the highest power of a prime number (x) in a factorial (N!), continuously divide N by x and add all the quotients.

Here it is calculated as 21/3 +7/3 +2/3 = 7+ 2+0 = 9.
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Re: What is the greatest integer p, for which 3^p is a factor of 21! ?  [#permalink]

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gmatbusters wrote:
To find the highest power of a prime number (x) in a factorial (N!), continuously divide N by x and add all the quotients.

Here it is calculated as 21/3 +7/3 +2/3 = 7+ 2+0 = 9.
Nice approach

Sent from my iPhone using GMAT Club Forum mobile app Re: What is the greatest integer p, for which 3^p is a factor of 21! ?   [#permalink] 02 Apr 2018, 15:42
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