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# What is the smallest positive integer that has exactly 18 positive fac

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Re: What is the smallest positive integer that has exactly 18 positive fac [#permalink]
Prime factorization of 18 =2×3×3
Therefore smallest number= 2^2×3^2×5=180

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Re: What is the smallest positive integer that has exactly 18 positive fac [#permalink]
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Bunuel wrote:
What is the smallest positive integer that has exactly 18 positive factors?

A. 180
B. 216
C. 240
D. 256
E. None of the above

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Let’s work the procedure backwards for finding the number of positive factors of a particular integer.

Let n be the smallest positive integer that has exactly 18 positive factors. Since 18 = 2 x 3 x 3 = (1 + 1)(2 + 1)(2 + 1), we want n = a^1 x b^2 x c^2, where a, b, and c are distinct prime numbers. Since we want n as small as possible, we want a, b, and c to be the three smallest prime numbers, i.e., 2, 3, and 5. However, to make n the smallest, we want the largest prime number to have the smallest exponent and the smallest prime number to have the largest exponent. Therefore, n = 5^1 x 3^2 x 2^2 = 5 x 9 x 4 = 180.

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Re: What is the smallest positive integer that has exactly 18 positive fac [#permalink]
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Re: What is the smallest positive integer that has exactly 18 positive fac [#permalink]
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