For any positive integer \(n,\) the length of \(n\) is defined as the number of prime factors whose product is \(n.\) For example, the length of \(75\) is \(3,\) since \(75 = 3\cdot 5\cdot 5.\) How many two-digit positive integers have length \(6?\)
A. None
B. One
C. Two
D. Three
E. Four
Answer: C
Source: GMAT Prep
For any positive integer \(n,\) the length of \(n\) is defined as the number of prime factors whose product is \(n.\)
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Let's first find the smallest value with length 6.M7MBA wrote: ↑Sun Sep 12, 2021 4:29 amFor any positive integer \(n,\) the length of \(n\) is defined as the number of prime factors whose product is \(n.\) For example, the length of \(75\) is \(3,\) since \(75 = 3\cdot 5\cdot 5.\) How many two-digit positive integers have length \(6?\)
A. None
B. One
C. Two
D. Three
E. Four
Answer: C
Source: GMAT Prep
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 2, two-digit positive integers with length 6.
Answer: C
Cheers,
Brent