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

BTGmoderatorDC wrote: ↑

Sun Jan 09, 2022 8:36 pm

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 * 5 * 5. How many two-digit positive integers have length 6?

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


OA C

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 2 x 2 x 2 x 2 x 2 x 2 = 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 x 2 x 2 x 2 x 2 x 2 = 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 3 x 3 x 2 x 2 x 2 x 2 = 144. This is a 3-digit positive integer. NO GOOD

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

Answer: C

Cheers,
Brent