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# What is the smallest positive integer n for which n!/18^8 is an intege

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Manager
Joined: 29 Oct 2019
Posts: 216
What is the smallest positive integer n for which n!/18^8 is an intege  [#permalink]

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03 Feb 2020, 10:57
1
8
00:00

Difficulty:

55% (hard)

Question Stats:

65% (02:13) correct 35% (02:27) wrong based on 63 sessions

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What is the smallest positive integer n for which n!/18^8 is an integer?

A. 33

B. 36

C. 48

D. 72

E. 144
Director
Joined: 28 Jul 2016
Posts: 937
Location: India
Concentration: Finance, Human Resources
Schools: ISB '18 (D)
GPA: 3.97
WE: Project Management (Investment Banking)
What is the smallest positive integer n for which n!/18^8 is an intege  [#permalink]

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03 Feb 2020, 11:40
1
$$18^8 = 3^{16}*2^8$$
thus the number should atleast be divisible by 3^16
look for option
highest power of 3 dividing the numbers will be
a) 33! will be divisible by =$$\frac{33}{3}+\frac{33}{9}+\frac{33}{27} = 15$$ (nope)
b) 36! = $$\frac{36}{3}+\frac{36}{9}+\frac{36}{37 }$$= 12+4+1= 17 thus this will be divisible
hence B
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Re: What is the smallest positive integer n for which n!/18^8 is an intege  [#permalink]

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04 Feb 2020, 00:26
sjuniv32 wrote:
What is the smallest positive integer n for which n!/18^8 is an integer?

A. 33

B. 36

C. 48

D. 72

E. 144

Solution

• Prime factorization of $$18 = 2*3^2$$
o $$18^8 = 2^8*3^{16}$$
• For, $$\frac{n! }{18^8}$$ to be an integer, prime factorization of n! must contains power of $$3 ≥ 16$$
o If n! contains $$3^{16}$$ then power of 2 in n! must be greater than 16. So, we need not to be worried about the powers of 2.
Now, let us check the answer options:
• Option A. $$n= 33$$
o Powers of 3 in 33! $$= [\frac{33}{3}] + [\frac{33}{9}] + [\frac{33}{27}] = 11+3+1 = 15$$ which is less than 16.
 Here, [] represents the greatest integer function.
o So, A cannot be the answer.
• Option B. $$n = 36$$
o Powers of 3 in $$36! = [\frac{36}{3}] + [\frac{36}{9}] + [\frac{36}{27}] = 12+4+1 = 17$$ which is greater than 16.
 Here, [] represents the greatest integer function.
o Hence, B can be the answer.
Since all other options are greater than 36 and we have asked for smallest integer n, so we need not to check further options.
Thus, the correct answer is Option B.
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Manager
Joined: 29 Oct 2019
Posts: 216
Re: What is the smallest positive integer n for which n!/18^8 is an intege  [#permalink]

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28 Feb 2020, 18:13
sjuniv32 wrote:
What is the smallest positive integer n for which n!/18^8 is an integer?

A. 33

B. 36

C. 48

D. 72

E. 144

Hi
Can I have some other qick solutions to this tricky problem?
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Re: What is the smallest positive integer n for which n!/18^8 is an intege  [#permalink]

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28 Feb 2020, 19:46
2
sjuniv32 wrote:
sjuniv32 wrote:
What is the smallest positive integer n for which n!/18^8 is an integer?

A. 33

B. 36

C. 48

D. 72

E. 144

Hi
Can I have some other qick solutions to this tricky problem?

Re: What is the smallest positive integer n for which n!/18^8 is an intege   [#permalink] 28 Feb 2020, 19:46
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