For how many positive integer values of n is 3^n a factor of $15 factorial (15!)?

Answer 1

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.

Answer 2

multiples of 3 ... five between 1 and 15 ... 9 is 3 squared

Answer 3

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!.

Answer 4

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