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# If 100!/30^n is an integer, which of the following is the greatest

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Math Expert
Joined: 02 Sep 2009
Posts: 58435
If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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28 Apr 2016, 03:31
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If 100!/30^n is an integer, which of the following is the greatest possible value of n?

A. 28
B. 24
C. 20
D. 15
E. 10

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Joined: 04 Dec 2015
Posts: 743
Location: India
Concentration: Technology, Strategy
WE: Information Technology (Consulting)
If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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14 Jun 2017, 20:48
5
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Bunuel wrote:
If 100!/30^n is an integer, which of the following is the greatest possible value of n?

A. 28
B. 24
C. 20
D. 15
E. 10

Here's the explanation.

Find the prime factors of $$30^n$$
30 = 2 x 3 x 5.
Therefore $$30^n = 2^n$$ x $$3^n$$ x $$5^n$$

Formula to find powers of prime numbers present in $$n!$$ is $$= \frac{n}{p} + \frac{n}{p^2} + \frac{n}{p^3} .... + \frac{n}{p^k}$$ ----- (p is the prime factor. k must be such that $$p^k \geq{n}$$ )

Therefore we can find powers of prime factors (2, 3 and 5) in 100! by using the above formula.

Number of 2 in 100!
$$\frac{100}{2} + \frac{100}{4} + \frac{100}{8} + \frac{100}{16} + \frac{100}{32} + \frac{100}{64} = 50 + 25 + 12 + 6 + 3 + 1 = 97$$

Number of 3 in 100!
$$\frac{100}{3} + \frac{100}{9} + \frac{100}{27} + \frac{100}{81} = 33 + 11 + 3 + 1 = 48$$

Number of 5 in 100!
$$\frac{100}{5} + \frac{100}{25} = 20 + 4 = 24$$

Greatest value of power of n! would be the power of the prime factor which has the lowest power. In this case power of 5 which is 24.
Hence $$30^n = 30^{24}$$. Greatest value of n = 24. Answer B.

Hope its clear now.
##### General Discussion
Marshall & McDonough Moderator
Joined: 13 Apr 2015
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Re: If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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28 Apr 2016, 03:52
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1
30^n = 2^n * 3^n * 5^n

100! has 97 2's, 48 3's and 24 5's

Since each of 2, 3 and 5 is required to form 30 and there are only 24 5's there can only be 24 30's in 100!

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Re: If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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14 Jun 2017, 23:16
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30= 2*3*5
The value of n depends on 5 since it the greatest divisor.
100/5=20; 100/25=4
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Re: If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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15 Oct 2018, 14:48
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Re: If 100!/30^n is an integer, which of the following is the greatest   [#permalink] 15 Oct 2018, 14:48
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