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

Answer: C

Source: GMAT Prep

## 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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Let's first find the smallest value with length 6.M7MBA wrote: ↑Sun Sep 12, 2021 4:29 amFor 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

Answer: C

Source: GMAT Prep

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

**3**x2x2x2x2x2 = 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 3x

**3**x2x2x2x2 = 144. This is a 3-digit positive integer. NO GOOD

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

Answer: C

Cheers,

Brent