## For any positive integer $$n,$$ the length of $$n$$ is defined as the number of prime factors whose product is $$n.$$

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### For any positive integer $$n,$$ the length of $$n$$ is defined as the number of prime factors whose product is $$n.$$

by M7MBA » Sun Sep 12, 2021 4:29 am

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For any positive integer $$n,$$ the length of $$n$$ is defined as the number of prime factors whose product is $$n.$$ For example, the length of $$75$$ is $$3,$$ since $$75 = 3\cdot 5\cdot 5.$$ How many two-digit positive integers have length $$6?$$

A. None
B. One
C. Two
D. Three
E. Four

Source: GMAT Prep

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### Re: For any positive integer $$n,$$ the length of $$n$$ is defined as the number of prime factors whose product is $$n.\ by [email protected] » Sun Sep 12, 2021 8:10 am M7MBA wrote: Sun Sep 12, 2021 4:29 am For any positive integer \(n,$$ the length of $$n$$ is defined as the number of prime factors whose product is $$n.$$ For example, the length of $$75$$ is $$3,$$ since $$75 = 3\cdot 5\cdot 5.$$ How many two-digit positive integers have length $$6?$$

A. None
B. One
C. Two
D. Three
E. Four

Source: GMAT Prep
Let's first find the smallest value with length 6.
This is the case when each prime factor is 2.
We get 2x2x2x2x2x2 = 64. This is a 2-digit positive integer. PERFECT

To find the next largest number with length 6, we'll replace one 2 with a 3
We get 3x2x2x2x2x2 = 96. This is a 2-digit positive integer. PERFECT

To find the third largest number with length 6, we'll replace another 2 with a 3
We get 3x3x2x2x2x2 = 144. This is a 3-digit positive integer. NO GOOD

So there are only 2, two-digit positive integers with length 6.