GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2508
Given Kudos: 1053
GPA: 3.39
For any positive integer n, the length of n is defined as the number o
[#permalink]
10 Dec 2022, 02:15
10 Dec 2022, 02:15
1
Expert Reply
00:00
Question Stats:
57% (02:23) correct
42% (01:59) wrong based on 7 sessions
Hide Show timer Statistics
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
[#permalink]
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
Moderators:
