math

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

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  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.

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

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  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!.

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  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

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