prime-saturated

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eureka123: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Sat Apr 09, 2011 6:20 am

prime-saturated

by eureka123 » Sun Apr 10, 2011 6:52 pm

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

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Source: Beat The GMAT — Problem Solving | Sun Apr 10, 2011 6:52 pm

HSPA: Legendary Member; Posts: 1101; Joined: Fri Jan 28, 2011 7:26 am; Thanked: 47 times; Followed by:13 members; GMAT Score:640

by HSPA » Sun Apr 10, 2011 7:07 pm

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

First take: 640 (50M, 27V) - RC needs 300% improvement
Second take: coming soon..
Regards,
HSPA.

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Tani: Legendary Member; Posts: 1255; Joined: Fri Nov 07, 2008 2:08 pm; Location: St. Louis; Thanked: 312 times; Followed by:90 members

by Tani » Sun Apr 10, 2011 7:27 pm

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.

Tani Wolff

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Sun Apr 10, 2011 7:28 pm

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)!