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

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

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New post 28 Apr 2016, 02:31
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A
B
C
D
E

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

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New post 14 Jun 2017, 19:48
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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. :)
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Re: If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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New post 28 Apr 2016, 02:52
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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!

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

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New post 14 Jun 2017, 22:16
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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
Hence n=24; Answer B
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Re: If 100!/30^n is an integer, which of the following is the greatest  [#permalink]

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New post 15 Oct 2018, 13:48
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Re: If 100!/30^n is an integer, which of the following is the greatest   [#permalink] 15 Oct 2018, 13:48
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