Events & Promotions
|
|
GMAT 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.
Apr 2610:00 AM EDT
-11:00 AM EDT
The Target Test Prep course represents a quantum leap forward in GMAT preparation, a radical reinterpretation of the way that students should study. Try before you buy with a 5-day, full-access trial of the course for FREE!
Apr 2601:30 AM EDT
-02:30 AM EDT
Learn how Keshav, a Chartered Accountant, scored an impressive 705 on GMAT in just 30 days with GMATWhiz's expert guidance. In this video, he shares preparation tips and strategies that worked for him, including the mock, time management, and more.
Apr 2711:00 AM EDT
-12:00 PM EDT
Prefer video-based learning? The Target Test Prep OnDemand course is a one-of-a-kind video masterclass featuring 400 hours of lecture-style teaching by Scott Woodbury-Stewart, founder of Target Test Prep and one of the most accomplished GMAT instructors
Apr 2808:00 AM PDT
-11:00 AM PDT
Whether you are just beginning your MBA journey or fine-tuning your path to a 700+ score, GMAT Day is designed to give you a competitive edge.
May 0111:00 AM PDT
-12:00 PM PDT
Free- full-length test + 15 concept videos + 150+ short videos + study plan!
28 Apr 2017, 02:57
Kudos
Bookmarks
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
36% (02:07) correct
64%
(01:55)
wrong
based on 301
sessions
History
Date
Time
Result
Not Attempted Yet
A positive integer is called semiprime if it is the product of exactly two not-necessarily-distinct prime numbers. A positive integer is called highly composite if it has more factors than any smaller positive integer has. How many positive integers are both semiprime and highly composite?
A. 0
B. 1
C. 2
D. 3
E. Infinitely many
A. 0
B. 1
C. 2
D. 3
E. Infinitely many
28 Apr 2017, 09:17
Kudos
Bookmarks
Examples of Semi-Primes
4 = 2x2
6 = 2x3
10 = 2x5
The smallest Semi Prime number is 4 which has 3 factors. The numbers smaller than 4 are 1,2 & 3. All three have 1,2,& 2 factors respectively. Which means 4 has higher number of factors than the number smiler than it, hence 4 is a semi-prime highly composite number.
The next Semi-Prime is 6 which has 4 factors. The numbers smaller than 5 are 1,2,3,4 & 5. All give have 1,2,2,3 & 2 factors respectively. Which means 6 has higher number of factors than the number smiler than it, hence 6 is a semi-prime highly composite number.
The next Semi-Prime is 10 which has 4 factors. The numbers smaller than 10 do not have factors less than 4. Hence 10 is not a semi-prime highly composite number.
This brings to the point that a semi-prime number if it is a square, will have 3 factors and it the semi-prime number is not a square will have 4 factors. As we can see above that semi-primes after 6 will either have 3 or 4 factors and numbers smaller than them will either than 3 or 4 factors, so semi-primes after 6 will not be a semi prime highly composite. Hence we have only 2 semi prime highly composite: 4 & 6.
Answer is C. 2
4 = 2x2
6 = 2x3
10 = 2x5
The smallest Semi Prime number is 4 which has 3 factors. The numbers smaller than 4 are 1,2 & 3. All three have 1,2,& 2 factors respectively. Which means 4 has higher number of factors than the number smiler than it, hence 4 is a semi-prime highly composite number.
The next Semi-Prime is 6 which has 4 factors. The numbers smaller than 5 are 1,2,3,4 & 5. All give have 1,2,2,3 & 2 factors respectively. Which means 6 has higher number of factors than the number smiler than it, hence 6 is a semi-prime highly composite number.
The next Semi-Prime is 10 which has 4 factors. The numbers smaller than 10 do not have factors less than 4. Hence 10 is not a semi-prime highly composite number.
This brings to the point that a semi-prime number if it is a square, will have 3 factors and it the semi-prime number is not a square will have 4 factors. As we can see above that semi-primes after 6 will either have 3 or 4 factors and numbers smaller than them will either than 3 or 4 factors, so semi-primes after 6 will not be a semi prime highly composite. Hence we have only 2 semi prime highly composite: 4 & 6.
Answer is C. 2
General Discussion
08 Jul 2020, 08:53
Kudos
Bookmarks
OFFICIAL SOLUTION
Since the number in question is semiprime, it has exactly two prime factors. There are only two cases to consider: The case in which the prime factors are the same, and the case in which they are different.
Consider the case in which the two prime factors are the same. Since we're looking for the smallest possible such positive integer (in order for the number to be highly composite), take the prime factors as 2
and 2
. The number in question will be 2∗2=4
, and it is easy to confirm that 4
has more factors (three: 1
, 2
, and 4
) than does any smaller positive integer. Any other square of a prime will still have just three total factors but will be larger than 4
and so will not be highly composite.
Now consider the case in which the two prime factors are different. Since we're still looking for the smallest possible such positive integer (in order for the number to be highly composite), take the prime factors as 2
and 3
. The number in question will be 2∗3=6
, and it is easy to confirm that 6
has more factors (four: 1
, 2
, 3
, and 6
) than does any smaller positive integer. Any other product of two distinct primes will still have exactly four total factors but will be larger than 6
and so will not be highly composite.
Thus 4
and 6
are the only two numbers that are both semiprime and highly composite. C is correct.
Since the number in question is semiprime, it has exactly two prime factors. There are only two cases to consider: The case in which the prime factors are the same, and the case in which they are different.
Consider the case in which the two prime factors are the same. Since we're looking for the smallest possible such positive integer (in order for the number to be highly composite), take the prime factors as 2
and 2
. The number in question will be 2∗2=4
, and it is easy to confirm that 4
has more factors (three: 1
, 2
, and 4
) than does any smaller positive integer. Any other square of a prime will still have just three total factors but will be larger than 4
and so will not be highly composite.
Now consider the case in which the two prime factors are different. Since we're still looking for the smallest possible such positive integer (in order for the number to be highly composite), take the prime factors as 2
and 3
. The number in question will be 2∗3=6
, and it is easy to confirm that 6
has more factors (four: 1
, 2
, 3
, and 6
) than does any smaller positive integer. Any other product of two distinct primes will still have exactly four total factors but will be larger than 6
and so will not be highly composite.
Thus 4
and 6
are the only two numbers that are both semiprime and highly composite. C is correct.
