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# What is the greatest value of n such that 18^n is a factor of 18! ?

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Math Expert
Joined: 02 Sep 2009
Posts: 58442
What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 00:20
1
10
00:00

Difficulty:

65% (hard)

Question Stats:

49% (01:31) correct 51% (01:18) wrong based on 83 sessions

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What is the greatest value of n such that 18^n is a factor of 18! ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 00:42
1
18/3 + 18/3^2 = 6+2=8
But 18=2*3^2.
Hence n=8/2=4.

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 03:41
1
Bunuel wrote:
What is the greatest value of n such that 18^n is a factor of 18! ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Project PS Butler

18! = 1*2*3*4......17*18

$$18 = 2*3^2$$

i.e. to find the power of 18 we need to find the powers of 3 in 18!

Power of 3 in 18! = [$$18/3$$] + [$$18/3^2$$] + [$$18/3^3$$] = 6+2+0 = 8

i.e. $$18! = 2^16*3^8*$$.. $$= (18)^4*2^12$$*...

i.e. n = 4

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 06:07
I am not sure but I answered C.

I took the factors of 18! and tried to make as many 18 using different combination. I have:

18 x 1
9 x 2
6 x 3

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What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 06:35
azl wrote:
I am not sure but I answered C.

I took the factors of 18! and tried to make as many 18 using different combination. I have:

18 x 1
9 x 2
6 x 3

The 3s from 12, 15 multiplied by any 2 will also make an 18!

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 09:04
TakingThePlunge wrote:
azl wrote:
I am not sure but I answered C.

I took the factors of 18! and tried to make as many 18 using different combination. I have:

18 x 1
9 x 2
6 x 3

The 3s from 12, 15 multiplied by any 2 will also make an 18!

Posted from my mobile device

Oh! Well done! I haven't seen that one.

Thanks for that !
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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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29 Aug 2019, 09:36
1
Bunuel wrote:
What is the greatest value of n such that 18^n is a factor of 18! ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Project PS Butler

Asked: What is the greatest value of n such that 18^n is a factor of 18! ?

18 = 2*3^2

Highest power of 3 in 18! = 6 + 2 = 8
Highest power of 2 in 18! = 9 + 4 + 2 + 1 = 16

18! = 2^16 * 3^8 * k
18! = 2^4*3^8 * 2^12k = (2*3^2)^4 * 2^12k

Greatest value of n = 4

IMO D

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?  [#permalink]

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02 Sep 2019, 19:38
Bunuel wrote:
What is the greatest value of n such that 18^n is a factor of 18! ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Project PS Butler

Since 18 = 3^2 x 2^1, each factor of 18 must contain two 3’s and one 2. Since there are fewer 3^2’s in 18! than 2’s, we simplify can determine the number of 3’s in 18!.

There is 1 three in 3, 1 three in 6, 2 three’s in 9, 1 three in 12, 1 three in 15, and 2 threes in 18. Thus, we have a total of 8 threes, so there is a total of four 3^2’s in 18!.

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Re: What is the greatest value of n such that 18^n is a factor of 18! ?   [#permalink] 02 Sep 2019, 19:38
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