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28 Apr 2016, 03:31
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
65% (01:19) correct
35%
(01:54)
wrong
based on 350
sessions
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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
A. 28
B. 24
C. 20
D. 15
E. 10
sashiim20
Joined: 04 Dec 2015
Last visit: 05 Jun 2024
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Location: India
Concentration: Technology, Strategy
Schools: ISB '19 IIMB IIMA XLRI HEC Sept19 intake Rotman '21 NUS '21 NTU '20 Trinity MBA '20 Bocconi '21 Smurfit "21
WE:Information Technology (Consulting)
Schools: ISB '19 IIMB IIMA XLRI HEC Sept19 intake Rotman '21 NUS '21 NTU '20 Trinity MBA '20 Bocconi '21 Smurfit "21
Posts: 608
14 Jun 2017, 20:48
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Bunuel
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
28 Apr 2016, 03:52
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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
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
