For how many positive integer values of n is 3^n a factor of $15 factorial (15!)?
We first determine the largest positive integer value of n such that 3^n | 15!. We determine this by counting the number of factors of 3 in the product. There are 5 multiples of 3 in the product, and there is one extra factor of 3 in 9, so there are a total of 5 + 1 = 6 factors of 3 in the product of the first 15 integers. So, for all n between 1 and 6, inclusive, 3^n is a factor of 15!
so it is 6
To find the number of positive integer values of n for which 3^n is a factor of 15!, we need to determine the exponent of the prime factor 3 in the prime factorization of 15!.
The exponent of a prime number p in the prime factorization of n! can be found using the formula:
exponent of p = [n/p] + [n/p^2] + [n/p^3] + ...
where [x] represents the greatest integer function and p^k denotes the kth power of prime p.
In this case, we need to find the exponent of 3 in the prime factorization of 15!. Since 3 is a prime number, we only need to consider values of n where 3^k ≤ 15.
Let's evaluate the formula for 3 and find the exponent:
Exponent of 3 = [15/3] + [15/3^2] + [15/3^3] + ...
= 5 + 1 + 0 + ...
= 6
Therefore, the exponent of 3 in the prime factorization of 15! is 6. This means that 3^6 is the highest power of 3 that divides 15!.
Since 3^n is a factor of 15! when n is a positive integer and n ≤ 6, there are a total of 6 positive integer values of n for which 3^n is a factor of 15!.
We first determine the largest positive integer value of $n$ such that $3^n | 15!$. We determine this by counting the number of factors of 3 in the product. There are 5 multiples of 3 in the product, and there is one extra factor of 3 in 9, so there are a total of $5+ 1 = \boxed{6}$ factors of 3 in the product of the first 15 integers. So, for all $n$ between 1 and 6, inclusive, $3^n$ is a factor of 15!.
multiples of 3 ... five between 1 and 15 ... 9 is 3 squared
15!
= (3x5)(14)13)(3x4)(10)(3x3)(8)(7)(3x2)(5)(4)(3)(2)(1)
I count 5 3's
so factors which are powers of 3 are
3^1, 3^2, 3^3, 3^4, and 3^5, so n can be 5 different integers.