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

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

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


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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.

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

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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