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GRE Prep Club Team Member
Joined: 20 Feb 2017
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For any positive integer n, the length of n is defined as the number o
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10 Dec 2022, 02:15
10 Dec 2022, 02:15
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Question Stats:
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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 * 5 * 5. How many two-digit positive integers have length 6?
A. None
B. One
C. Two
D. Three
E. Four
A. None
B. One
C. Two
D. Three
E. Four
Re: For any positive integer n, the length of n is defined as the number o
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11 Dec 2022, 14:03
11 Dec 2022, 14:03
1
Expert Reply
Basically the length of the integer is the sum of the powers of its prime factors.
Length of six means that the sum of the powers of primes of the integer (two digit) must be \(6\). First we can conclude that \(5\) can not be a factor of this integer as the smallest integer with the length of six that has \(5\) as prime factor is \(2^5*5=160\) (length=5+1=6), not a two digit integer.
The above means that the primes of the two digit integers we are looking for can be only \(2\) and/or \(3\). \(n=2^p*3^q\), \(p+q=6\) max value of \(p\) and \(q\) is \(6\).
Let's start with the highest value of \(p\):
\(n=2^6*3^0=64\) (length=6+0=6);
\(n=2^5*3^1=96\) (length=5+1=6);
\(n=2^4*3^2=144\) (length=4+2=6) not good as 144 is a three digit integer.
With this approach we see that actually \(5<=p<=6\).
Answer: C.
Length of six means that the sum of the powers of primes of the integer (two digit) must be \(6\). First we can conclude that \(5\) can not be a factor of this integer as the smallest integer with the length of six that has \(5\) as prime factor is \(2^5*5=160\) (length=5+1=6), not a two digit integer.
The above means that the primes of the two digit integers we are looking for can be only \(2\) and/or \(3\). \(n=2^p*3^q\), \(p+q=6\) max value of \(p\) and \(q\) is \(6\).
Let's start with the highest value of \(p\):
\(n=2^6*3^0=64\) (length=6+0=6);
\(n=2^5*3^1=96\) (length=5+1=6);
\(n=2^4*3^2=144\) (length=4+2=6) not good as 144 is a three digit integer.
With this approach we see that actually \(5<=p<=6\).
Answer: C.
gmatclubot
Re: For any positive integer n, the length of n is defined as the number o [#permalink]
11 Dec 2022, 14:03
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