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Math Expert V
Joined: 02 Sep 2009
Posts: 59712
What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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1
9 00:00

Difficulty:   25% (medium)

Question Stats: 64% (00:55) correct 36% (00:59) wrong based on 256 sessions

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What is the greatest value of x such that 9^x is a factor of 9! ?

A. 1
B. 2
C. 3
D. 4
E. 5

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Joined: 05 Mar 2015
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What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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Bunuel wrote:
What is the greatest value of x such that 9^x is a factor of 9! ?

A. 1
B. 2
C. 3
D. 4
E. 5

9^x==3^(2x)
9/3=3
9/3^2=1

total=3+1=4
thus 2x=4 (3^4=3^2x)
means x=2

Ans B
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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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2
Bunuel wrote:
What is the greatest value of x such that 9^x is a factor of 9! ?

A. 1
B. 2
C. 3
D. 4
E. 5

$$9 = 3^2$$

9! has four 3s

9/3 = 3
3/1 = 1

So, There are two 9's in 9!

Hence, answer will be (B) 2

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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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Bunuel wrote:
What is the greatest value of x such that 9^x is a factor of 9! ?

A. 1
B. 2
C. 3
D. 4
E. 5

let's rewrite it as:
1*2*3*4*5*6*7*8*9/9^x
x can only be 2.

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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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1
incorrect tag for distance/rate prob
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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warriorguy wrote:
incorrect tag for distance/rate prob

Edited. Thank you for noticing.
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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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Can someone explain how there are two 9s in 9!?

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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mcmoorthy wrote:
Can someone explain how there are two 9s in 9!?

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app

9! = 1*2*3*4*5*6*7*8*9 = 1*2*3*4*5*(2*3)*7*8*9

3 and 3 (in 6) give one more 9.

Hope it's clear.
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Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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Bunuel wrote:
mcmoorthy wrote:
Can someone explain how there are two 9s in 9!?

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app

9! = 1*2*3*4*5*6*7*8*9 = 1*2*3*4*5*(2*3)*7*8*9

3 and 3 (in 6) give one more 9.

Hope it's clear.

For more check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.

Hope this helps.
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Target Test Prep Representative V
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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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Bunuel wrote:
What is the greatest value of x such that 9^x is a factor of 9! ?

A. 1
B. 2
C. 3
D. 4
E. 5

We need to determine the greatest values of x such that 9^x (or 3^2x) divides into 9!. In other words, we need to determine the number of factors of 9 in 9!. Since 9 = 3^2, let’s first determine the total number of factors of 3 in 9!.

To do so, we can use the following shortcut in which we divide 9 by 3, then divide the quotient (ignore any nonzero remainder) by 3 and continue this process until we no longer get a nonzero quotient.

9/3 = 3

3/3 =1

Since 1/3 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 9!.

Thus, there are 3 + 1 = 4 factors of 3 within 9!, and therefore there are 2 factors of 9 in 9!.

Alternate solution:

Recall that 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. We see that the first factor 9 is a factor of 9. We also see that 6 x 3 = 18 will contain another factor of 9. Thus there are only 2 factors of 9 in 9! since none of the other numbers contain or contribute a factor of 9 or 3. Thus, the largest value of x such that 9^x is a factor of 9! is 2.

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Re: What is the greatest value of x such that 9^x is a factor of 9! ?  [#permalink]

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It's an easy problem if you know the concept.

Prime factorization of 9 = 3^2

9/3 + 9/9 = 4

=> it appears twice

Ans B Re: What is the greatest value of x such that 9^x is a factor of 9! ?   [#permalink] 20 Nov 2019, 05:53
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