Problem Solving

Could some one explain the concept of Prime- saturated?
Thanks,


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?
A. 99
B. 98
C. 97
D. 96
E. 95

I think 96 has prime factors of 2 and 3.. whose product is 6 < 9

IF the product has to be less than the square root of "n", with any of these options, the product of the distinct prime factors has to be less than 10. Since we are dealing with integers that means the product is < or = 9.

Looking at primes, the only two distinct primes whose product is less than ten would be 2*3. Therefore the answer can only have 2 and 3 as prime factors. And yes, the answer is D.

eureka123 wrote:Could some one explain the concept of Prime- saturated?
Thanks,


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?
A. 99
B. 98
C. 97
D. 96
E. 95

Hi,

you're not expected to be familiar with that term before seeing this question - the question provides the definition.

According to the definition, "prime saturated" means that the product of the distinct prime factors must be less than the root. Once we understand the definition, we go to the choices. We note that all of the choices are between 81 and 100, so they all have roots between 9 and 10.

So, rephrasing the question, keeping the choices in mind:

What's the biggest one of these numbers for which the product of the primes is less than 10?

99 = 3*3*11

Is 3*11 < 10? Nope!

98 = 2*49 = 2*7*7

is 2*7 < 10? Nope!

97 = a prime.

Is 97 < 10? Nope!

96 = 3*32 = 3*2^5

Is 3*2 < 10? Yes - choose (D)!

Thanks a lot to All. Very good explanation.