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# What is the greatest value of q such that 9^q is a factor of 21! ?

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Math Expert
Joined: 02 Sep 2009
Posts: 46297
What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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13 Dec 2016, 10:52
00:00

Difficulty:

45% (medium)

Question Stats:

58% (01:08) correct 42% (01:19) wrong based on 193 sessions

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What is the greatest value of q such that 9^q is a factor of 21! ?

A. 1
B. 3
C. 4
D. 5
E. 6

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Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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13 Dec 2016, 13:15
4
$$9 = 3^2$$

First we need to find max power of $$3$$ in $$21!$$

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

But we are asked about max power of 9 not 3, so we need to divide our power of 3 by 2 and take integer value.

$$[\frac{9}{2}] = 4$$

$$q=4$$

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Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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13 Dec 2016, 14:08
1
First find the number of 3s in 21! when prime factoring each term. E.g. how many powers of 3 in 21!.

21 -3*7
20 - NA
19 - NA
18 - 2*(3^2)
17 - NA
16 - NA
15 - 5*3
14 - NA
13 - NA
12 - (2^2)*3
11 - NA
10 - NA
9 - 3^2
8 - NA
7 - NA
6 - 2*3
5- NA
4 - NA
3 - 3^1
2 - NA
1 - NA

Totalling up the number of 3s gives 3^9 = (3^2)^4.5 = 9^4.5. Therefore the highest power of 9 must be 4 (5 is larger than 4.5).
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Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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14 Dec 2016, 11:08
1
Bunuel wrote:
What is the greatest value of q such that 9^q is a factor of 21! ?

A. 1
B. 3
C. 4
D. 5
E. 6

$$9 = 3^2$$

HIghest power of 3 in 21! is 9

21/3 = 7
7/3 = 2

So, the highest power of 9 will be 1/2*9 => 4

Hence, correct answer will be (C) 4

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Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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16 Dec 2016, 10:34
1
1
Bunuel wrote:
What is the greatest value of q such that 9^q is a factor of 21! ?

A. 1
B. 3
C. 4
D. 5
E. 6

9=3^2
largest power of 3 in 21! is 21/3 + 21/9 = 7+2=9
9^q= 3^9
q=9/2 = 4
C
Intern
Joined: 03 Apr 2016
Posts: 16
Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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16 Dec 2016, 11:38
2
http://www.campusgate.co.in/2011/10/fin ... e.html?m=1

Sent from my SAMSUNG-SM-G935A using GMAT Club Forum mobile app
Senior Manager
Joined: 05 Dec 2016
Posts: 260
Concentration: Strategy, Finance
GMAT 1: 620 Q46 V29
Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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16 Dec 2016, 13:24
1
9^q=3^2q
To find the answer we it is needed to find the number of powers of 3 in 21!, that equals to 21/3^1 = 7; 21/3^2=2
7+2=9 number of powers of 2 in 21!
Since initial condition is that 3^2q, 9=2q; q=9/2=4,5, therefore maximum power of 3 in 21! is 4. Answer C is correct.
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Re: What is the greatest value of q such that 9^q is a factor of 21! ? [#permalink]

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15 Sep 2017, 11:18
1
http://www.campusgate.co.in/2011/10/fin ... e.html?m=1

Sent from my SAMSUNG-SM-G935A using GMAT Club Forum mobile app

Thanks for the link, it has helped me to solve my conceptual gap.
Re: What is the greatest value of q such that 9^q is a factor of 21! ?   [#permalink] 15 Sep 2017, 11:18
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