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n is a product of 6 distinct prime numbers. m!/n is an integ

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by Max@Math Revolution » Fri Jan 11, 2019 3:33 am

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[Math Revolution GMAT math practice question]

n is a product of 6 distinct prime numbers. m!/n is an integer.
What is the smallest possible value of m?

A. 10
B. 11
C. 12
D. 13
E. 14
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by fskilnik@GMATH » Fri Jan 11, 2019 5:54 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

n is a product of 6 distinct prime numbers. m!/n is an integer.
What is the smallest possible value of m?

A. 10
B. 11
C. 12
D. 13
E. 14
$$\left. \matrix{
n = {p_1} \cdot {p_2} \cdot \ldots \cdot {p_6}\,\,\,{\rm{primes}}\,\,{\rm{and}}\,\,{\rm{distincts}}\,\,\, \hfill \cr
{{m!} \over n} = {\mathop{\rm int}} \hfill \cr
? = m\,\,\min \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{Take}}\,\,\left( {{p_1},{p_2},{p_3},{p_4},{p_5},{p_6}} \right) = \left( {2,3,5,7,11,13} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 13$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by swerve » Sun Jan 13, 2019 8:14 am
Smallest 6 prime numbers = 2, 3, 5, 7, 11, 13

So, m! should include all of them.

Implies smallest possible value of m = 13

Therefore, the correct answer is D
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by Max@Math Revolution » Sun Jan 13, 2019 4:39 pm
=>

The smallest product of 6 distinct prime numbers is n = 2*3*5*7*11*13. 13! Is the smallest factorial that is divisible by 2*3*5*7*11*13. Thus, the smallest possible integer value of m is 13.
Therefore, the answer is D.
Answer: D
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by Scott@TargetTestPrep » Mon Jan 14, 2019 5:53 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

n is a product of 6 distinct prime numbers. m!/n is an integer.
What is the smallest possible value of m?

A. 10
B. 11
C. 12
D. 13
E. 14
The smallest product of 6 distinct prime numbers is:

2 x 3 x 5 x 7 x 11 x 13, so we see that m! must have a prime of 13, so the least value of m is 13.

Answer: D

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