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What is the greatest value...

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by BTGmoderatorLU » Thu Oct 19, 2017 6:53 pm
What is the greatest value of integer n such that 5^n is a factor of 15! ?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Can any expert explain this PS question please? I don't have it clear. Thanks.
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by GMATGuruNY » Fri Oct 20, 2017 4:47 am
LUANDATO wrote:What is the greatest value of integer n such that 5^n is a factor of 15! ?

A. 1
B. 2
C. 3
D. 4
E. 5
Since 5^n must be a factor of 15!, we need to count the number of 5's that can divide into 15!.
Within 15! are the following multiples of 5:
15 = 3*5
10 = 2*5
5 = 1*5.

As illustrated by the blue values above, the prime-factorization of 15! includes exactly three 5's.
Thus, at most three 5's can divide into 15!, with the result that the greatest possible value for n is 3.

The correct answer is C.

A similar problem:
https://www.beatthegmat.com/integers-t270158.html
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by Scott@TargetTestPrep » Fri Nov 22, 2019 11:53 am
BTGmoderatorLU wrote:What is the greatest value of integer n such that 5^n is a factor of 15! ?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Can any expert explain this PS question please? I don't have it clear. Thanks.
Since 15! has 3 prime factors of 5 (1 prime factor of 5 from each of the numbers 5, 10 and 15), the greatest value of n is 3.

Answer: C

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by Brent@GMATPrepNow » Fri Nov 22, 2019 3:49 pm
BTGmoderatorLU wrote:What is the greatest value of integer n such that 5^n is a factor of 15! ?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Can any expert explain this PS question please? I don't have it clear. Thanks.
-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:

If k is a factor of N, then k is "hiding" within the prime factorization of N

Consider these examples:
3 is a factor of 24, because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a factor of 70 because 70 = (2)(5)(7)
And 8 is a factor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a factor of 630 because 630 = (2)(3)(3)(5)(7)
-----BACK TO THE QUESTION!---------------------

What is the greatest value of integer n such that 5^n is a factor of 15! ?
In other words, "How many 5's are hiding within the prime factorization of 15!"
15! = (15)(14)(13)(12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)
= (3)(5)(14)(13)(12)(11)(2)(5)(9)(8)(7)(6)(5)(4)(3)(2)(1)
There are three 5's in the prime factorization of 15!
In other words, 5³ is a factor of 15!

Answer: C

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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