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

##### This topic has expert replies
Moderator
Posts: 1873
Joined: 29 Oct 2017
Thanked: 1 times
Followed by:5 members

### 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

00:00

A

B

C

D

E

## Global Stats

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

### GMAT/MBA Expert

GMAT Instructor
Posts: 15787
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1267 members
GMAT Score:770

### 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.