What is the smallest positive integer n for which n!/9^9 is an integer

Question

What is the smallest positive integer n for which n!/9^9 is an integer?

A. 36
B. 39
C. 54
D. 78
E. 81

JeffTargetTestPrep · Accepted Answer

We need to determine the largest value of n such that n!/(9^9) is an integer. Let’s start by simplifying 9^9.

9^9 = (3^2)^9 = 3^18.

Thus, we need to find the smallest positive integer n such that n! contains at least 18 factors of 3 in its prime factorization.

Let’s start with answer choice A: 36.

We can determine the number of 3s in 36! by using the following shortcut in which we divide 36 by 3 then divide the quotient of 36/3 by 3, and continue this process until we no longer get a nonzero quotient.
 
36/3 = 12
 
12/3 = 4

4/3 = 1 (we can ignore the remainder)
 
Since 1/3 does not produce a nonzero quotient, we can stop.
 
The final step is to add up our quotients; that sum represents the number of factors of 3 within 36!.
 
Thus, there are 12 + 4 + 1 = 17 factors of 3 within 36!.

Looking at answer choice B, 39, we see that 39! factorial will have one more factor of 3 than 36! (because 39 has a prime factor of 3), and thus, 39! will have 18 factors of 3.

Thus, 39 is the minimum value of n.

Answer: B

0akshay0 · Answer

9^9 = 3^{18}

lets consider the smallest number from the given options i.e. 36
number of multiples of 3 in 36! = 36/3 + 36/9 + 36/27 = 12 + 4 + 1 =17 
so 39! will have 18 multiples of 3 and  36!/3^{18}  will surely be an Integer.

Hence Option B is correct.

giobas · Answer

9^9 = 3^18
so we must find which of the answers contain 18 multiples of 3.
36 has 12 multiples of 3 and we must not forger factors from 9,27 and 36.in total 36 has 17 factors of 3
thus 39 will have 18 factors of 3

answer is B

reena.phogat · Answer

Ans is B
3^18
Number of 3 in 39!
39/3=13
12/3=4
4/3=1
Hence
13+4+1= 18

Sent from my XT1068 using  GMAT Club Forum mobile app

BillyZ · Answer

Official solution from Veritas Prep.

This is a number theory question – note the term “positive integer” and the fact that the question is essentially asking about the divisibility of  n! . As such, we will begin by prime factoring  9^9  as  (3^2)^{9}=3^{18} . The real question is how far we have to go to find eighteen copies of 3.

The cop-out answer would be to simply take  n=18∗3=54 ;  54!  definitely contains at least eighteen factors of 3, since it contains  3∗1, 3∗2, 3∗3, …, 3∗18 . However,  54!  actually contains far more than just the eighteen 3s, since several factors contain “bonus” 3s. For instance,  54!  contains 9, which is  3^2 , and 27, which is  3^3 , and even 54 itself, which is  2∗3^3 . The correct answer must be less than 54.

At this point we could calculate the exact number of 3s in  54!  (it has twenty-six of them, as it turns out) and adjust from there, but it might be simpler to just try answers A and B, since one or the other must be correct.

36!  contains twelve multiples of 3 ( 3∗1  through  3∗12 ), but it also contains four multiples of 9 ( 9∗1  through  9∗4 ), each of which provides an additional factor of 3, since  9=3^2 . And  36!  Even contains one multiple of  3^3=27 , which provides an “additional” additional 3. Taken together,  36!  contains  12+4+1=17  factors of 3, just short of our goal.

We can see at this point that  39!  provides exactly one more 3 than does  36! , since  39!=39∗38∗37∗36! .

Therefore,  39!  contains exactly 18 factors of 3, and answer B is correct.

Archit3110 · Answer

so as to get an integer value of n!/9^9 
the value of n! must have 9 nines in the nr . which is possible when we have a pair of 3's + multiple of 9's
using answer option
for n=39 ;39! 
pair of 9's ; 9,18,27,36 ; total 4 nines now we need 5 more or say 10 multiple of 3's
multiples of 3s i.e multiple of (3,6),(12,15),(21,24),(27,30),(33,39) ; (27 has 3*9) total 5 9's we have now
Option B 39 is correct

Kinshook · Answer

Asked: What is the smallest positive integer n for which n!/9^9 is an integer?

Highest power of 9 in n! = 9
Highest power of 3 in n! = 18

If n = 45; Highest power of 3 in 45! = 15 + 5 + 1 = 21
If n = 39; Highest power of 3 in 39! = 13 + 4 + 1 = 18

IMO B

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What is the smallest positive integer n for which n!/9^9 is an integer

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