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length of 75 is 3

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by sanju09 » Wed Apr 08, 2009 5:26 am
For any positive integer n, the length of n is defined as 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. 0
B. 1
C. 2
D. 3
E. 4


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by sacx » Wed Apr 08, 2009 5:51 am
Smallest prime number = 2

so the smallest number with length 6 would be 2*2*2*2*2*2 = 64

next prime number = 3, so the next number with length 6 = 2*2*2*2*2*3 = 96

now keep replacing 2 by 3 till we get a number greater than 99

2*2*2*2*3*3 = 144

so the only numbers with length 6 and less than 99 are 64 and 96
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