A positive integer n is said to be "prime-saturated" if the product of all the different positive prime factors of n is less than the square root of n. what is the greatest two-digit prime-saturated integer?

Answer: 96

Given the following about a positive integer, n:

n is "prime-saturated"

The product of all the different positive prime factors of n is less than the square root of n.

We are to find the greatest possible two-digit value of n

This question can only be solved by trial and error, starting with 99 (Since no options are given). We’ll test to see the highest number less than 100 that fit into the description of n.

If n = 99

Prime factors of 99 are 3 and 11

The product of 3 and 11 = 33

Square root of 99 = 9.95

33 < 9.95 is a false statement!

If n = 98

Prime factors of 98 are 2 and 7

The product of 2 and 7 is 14

Square root of 98 = 9.899

14 < 9.899 is a false statement!

If n = 97

97 itself is a prime number

A number cannot be lesser than it’s square root

97 < Sqrt (97) is obviously a false statement!

If n = 96

Prime factors of 96 are 2 and 3

The product of 2 and 3 is 6

Square root of 96 = 9.798

6 < 9.798 is a true statement!

Therefore, the greatest two-digit prime-saturated integer that fits into the description of n is 96