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by kkadvent » Fri May 10, 2013 7:52 am
How to solve these types of problem?

For any positive integer n is defined as the number of prime factor whose product is n. For example for 75 length is 3 because 75 = 3*5*5. How many two digit positive integer have length 6?
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by srcc25anu » Fri May 10, 2013 8:49 am
Smallest prime = 2
Greatest power of 2 that is a 2 digit number = 2^6 = 64
Next biggest prime = 3
Largest power of prime that is a 2 digit number = 3 ^ 4 = 81 but 81 cannot be combined with any other prime. Else it would give a 3 digit number.

So only possible numbers are 2^6 = 64 and 2^3 * 3 = 69
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by Atekihcan » Sat May 11, 2013 6:30 am
srcc25anu wrote:2^3 * 3 = 69
I guess you meant to write (2^5)*3 = 96 :)
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by Brent@GMATPrepNow » Sat May 11, 2013 6:44 am
Here's the full question:

For any positive integer n, the length n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75= 3 X 5 X 5. How many two-digit positive integers have length 6?

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


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 two, two-digit positive integers with length 6.

Answer = C

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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