Last visit was: 23 Apr 2026, 14:13 It is currently 23 Apr 2026, 14:13
Home
Messages
Tests
Schools
Events
Close
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

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
User avatar
efet
User avatar
Joined: 14 Jul 2010
Last visit: 18 Feb 2021
Posts: 64
Own Kudos:
91
 [79]
Given Kudos: 2
Posts: 64
Kudos: 91
 [79]
How many integers less than 100 have exactly 4 odd factors b Updated on: 27 May 2023, 02:35
Originally posted by efet on 17 Jul 2010, 11:53.
Last edited by Bunuel on 27 May 2023, 02:35, edited 2 times in total.
Edited the OA.
6
Kudos
Add Kudos
73
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
D
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:

33% (03:03) correct 67% (02:46) wrong based on 516 sessions

History

Date
Time
Result
Not Attempted Yet
How many positive integers less than 100 have exactly 4 odd factors but no even factors?

A. 11
B. 13
C. 15
D. 17
E. 19
ShowHide Answer
Official Answer
D
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,785
Own Kudos:
810,852
 [38]
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,785
Kudos: 810,852
 [38]
How many integers less than 100 have exactly 4 odd factors b 17 Jul 2010, 12:51
18
Kudos
Add Kudos
20
Bookmarks
Bookmark this Post
efet
how many integers less than 100 have exactly 4 odd factors but no even factors?

a. 11
b. 13
c. 15
d. 17
e. 19

--

Is 9 with exactly 4 odd factors
Show more

gurpreetsingh
this is only possible when factors are 1,a,b,c

where c is the number itself and c= a*b

when a = 3, b can be 5 7 9 11 13 17 19 23 29 31, total 10
when a = 5, b can be 7,11,13,17,19 , total 5
when a = 7, b can be 11 total 1

so total = 16 but its not there in the options.

PS: 9 does not have 4 factors...1,3,9 total 3 factors
Show more

Two cases are possible:

\(x^1y^1=xy=odd<100\), where \(x\) and \(y\) are odd prime numbers, then # of factors will be \((1+1)(1+1)=4\);
If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.

OR:
\(x^3=odd<100\), where \(x\) is odd prime number, then # of factors will be \((3+1)=4\).
x can be only 3 - 1 number.

Total \(9+5+2+1=17\).

Answer: D.

P.S. Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Hope it helps.
General Discussion
User avatar
gurpreetsingh
User avatar
Joined: 12 Oct 2009
Last visit: 15 Jun 2019
Posts: 2,266
Own Kudos:
3,962
 [4]
Given Kudos: 235
Status:<strong>Nothing comes easy: neither do I want.</strong>
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Products:
My Reviews
Schools: ISB '15 (M)
GMAT 2: 710 Q50 V35
Posts: 2,266
Kudos: 3,962
 [4]
How many integers less than 100 have exactly 4 odd factors b 17 Jul 2010, 12:08
4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
this is only possible when factors are 1,a,b,c

where c is the number itself and c= a*b

when a = 3, b can be 5 7 9 11 13 17 19 23 29 31, total 10
when a = 5, b can be 7,11,13,17,19 , total 5
when a = 7, b can be 11 total 1

so total = 16 but its not there in the options.

PS: 9 does not have 4 factors...1,3,9 total 3 factors
User avatar
gurpreetsingh
User avatar
Joined: 12 Oct 2009
Last visit: 15 Jun 2019
Posts: 2,266
Own Kudos:
3,962
Given Kudos: 235
Status:<strong>Nothing comes easy: neither do I want.</strong>
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Products:
My Reviews
Schools: ISB '15 (M)
GMAT 2: 710 Q50 V35
Posts: 2,266
Kudos: 3,962
How many integers less than 100 have exactly 4 odd factors b 17 Jul 2010, 12:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Thanks bunnel I missed 7,13 combination. I need to curb such mistakes. 99% of the time I miss the question because of such mistakes....:)
User avatar
efet
User avatar
Joined: 14 Jul 2010
Last visit: 18 Feb 2021
Posts: 64
Own Kudos:
91
 [1]
Given Kudos: 2
Posts: 64
Kudos: 91
 [1]
How many integers less than 100 have exactly 4 odd factors b 17 Jul 2010, 20:24
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I had this question in 800score's first test. QA is 15. Does this mean that they are wrong?
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,785
Own Kudos:
810,852
 [1]
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,785
Kudos: 810,852
 [1]
How many integers less than 100 have exactly 4 odd factors b 20 Jul 2010, 02:30
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
efet
I had this question in 800score's first test. QA is 15. Does this mean that they are wrong?
Show more

Integers less than 100 with exactly 4 odd factors but no even factors:

1. 15=3*5 --> factors 1, 3, 5, 15;
2. 21=3*7 --> factors 1, 3, 7, 21;
3. 27=3^3 --> factors 1, 3, 9, 27;
4. 33=3*11 --> factors 1, 3, 11, 33;
5. 35=5*7 --> factors 1, 5, 7, 35;
6. 39=3*13 --> factors 1, 3, 13, 39;
7. 51=3*17 --> factors 1, 3, 17, 51;
8. 55=5*11 --> factors 1, 5, 11, 55;
9. 57=3*19 --> factors 1, 3, 19, 57;
10. 65=5*13 --> factors 1, 5, 13, 65;
11. 69=3*23 --> factors 1, 3, 23, 69;
12. 77=7*11 --> factors 1, 7, 11, 77;
13. 85=5*17 --> factors 1, 5, 17, 85;
14. 87=3*29 --> factors 1, 3, 29, 87;
15. 91=7*13 --> factors 1, 7, 13, 91;
16. 93=3*31 --> factors 1, 3, 31, 93;
17. 95=5*19 --> factors 1, 5, 19, 95.

OA must be wrong.
User avatar
seekmba
User avatar
Joined: 17 Feb 2010
Last visit: 25 Sep 2014
Posts: 626
Own Kudos:
3,644
 [2]
Given Kudos: 6
Posts: 626
Kudos: 3,644
 [2]
How many integers less than 100 have exactly 4 odd factors b 20 Jul 2010, 10:58
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hey Bunuel,

Quote:
xy = odd<100, where x and y are odd prime numbers, then # of factors will be ;

If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.
Show more

I did not understand, why the number '3' in the first line was not counted. If '3' is also counted then the total numbers will be 10 for the first line, 6 for the second and 3 for the 3rs line. Please explain.

Also, is there any other easy way to get to the solution? Thinking and working out the solution in the manner that you showed seems very daunting to me.....it was like a genius solution. It might never occur to me to solve it this way on test day.......Or is it a matter of practice.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,785
Own Kudos:
810,852
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,785
Kudos: 810,852
How many integers less than 100 have exactly 4 odd factors b 20 Jul 2010, 11:58
Kudos
Add Kudos
Bookmarks
Bookmark this Post
seekmba
Hey Bunuel,

Quote:
xy = odd<100, where x and y are odd prime numbers, then # of factors will be ;

If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.

I did not understand, why the number '3' in the first line was not counted. If '3' is also counted then the total numbers will be 10 for the first line, 6 for the second and 3 for the 3rs line. Please explain.
Show more

If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers --> we are counting # of \(xy\) (not \(x\) and \(y\) separately) --> \(3*5\), \(3*7\), \(3*11\), ... so 9 numbers.

seekmba
Also, is there any other easy way to get to the solution? Thinking and working out the solution in the manner that you showed seems very daunting to me.....it was like a genius solution. It might never occur to me to solve it this way on test day.......Or is it a matter of practice.
Show more

First you should realize that the numbers we are looking for can be either of a type \(xy\) or \(x^3\) (where \(x\) and \(y\) are distinct primes), next count them: should take less than 2 minutes. You are right it's matter of practice.
User avatar
PathFinder007
User avatar
Joined: 10 Mar 2014
Last visit: 21 Oct 2018
Posts: 127
Own Kudos:
745
Given Kudos: 13
Posts: 127
Kudos: 745
How many integers less than 100 have exactly 4 odd factors b 07 Apr 2014, 04:48
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
efet
how many integers less than 100 have exactly 4 odd factors but no even factors?

a. 11
b. 13
c. 15
d. 17
e. 19

--

Is 9 with exactly 4 odd factors

gurpreetsingh
this is only possible when factors are 1,a,b,c

where c is the number itself and c= a*b

when a = 3, b can be 5 7 9 11 13 17 19 23 29 31, total 10
when a = 5, b can be 7,11,13,17,19 , total 5
when a = 7, b can be 11 total 1

so total = 16 but its not there in the options.

PS: 9 does not have 4 factors...1,3,9 total 3 factors

Two cases are possible:

\(x^1y^1=xy=odd<100\), where \(x\) and \(y\) are odd prime numbers, then # of factors will be \((1+1)(1+1)=4\);
If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.

OR:
\(x^3=odd<100\), where \(x\) is odd prime number, then # of factors will be \((3+1)=4\).
x can be only 3 - 1 number.

Total \(9+5+2+1=17\).

Answer: D.

P.S. Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Hope it helps.
Show more

Hi Bunnel

What you mean by following

OR:
x^3=odd<100, where x is odd prime number, then # of factors will be (3+1)=4.
x can be only 3 - 1 number.

Total 9+5+2+1=17.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,785
Own Kudos:
810,852
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,785
Kudos: 810,852
How many integers less than 100 have exactly 4 odd factors b 07 Apr 2014, 05:35
Kudos
Add Kudos
Bookmarks
Bookmark this Post
pawankumargadiya
Bunuel
efet
how many integers less than 100 have exactly 4 odd factors but no even factors?

a. 11
b. 13
c. 15
d. 17
e. 19

--

Is 9 with exactly 4 odd factors

gurpreetsingh
this is only possible when factors are 1,a,b,c

where c is the number itself and c= a*b

when a = 3, b can be 5 7 9 11 13 17 19 23 29 31, total 10
when a = 5, b can be 7,11,13,17,19 , total 5
when a = 7, b can be 11 total 1

so total = 16 but its not there in the options.

PS: 9 does not have 4 factors...1,3,9 total 3 factors

Two cases are possible:

\(x^1y^1=xy=odd<100\), where \(x\) and \(y\) are odd prime numbers, then # of factors will be \((1+1)(1+1)=4\);
If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.

OR:
\(x^3=odd<100\), where \(x\) is odd prime number, then # of factors will be \((3+1)=4\).
x can be only 3 - 1 number.

Total \(9+5+2+1=17\).

Answer: D.

P.S. Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Hope it helps.

Hi Bunnel

What you mean by following

OR:
x^3=odd<100, where x is odd prime number, then # of factors will be (3+1)=4.
x can be only 3 - 1 number.

Total 9+5+2+1=17.
Show more

(odd prime)^3 to be less than 100, (odd prime) can only be 3.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,785
Own Kudos:
810,852
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,785
Kudos: 810,852
How many integers less than 100 have exactly 4 odd factors b 26 Apr 2014, 09:37
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Similar question to practice: https://gmatclub.com/forum/how-many-of- ... 00708.html
User avatar
vitaliyGMAT
User avatar
Joined: 13 Oct 2016
Last visit: 26 Jul 2017
Posts: 297
Own Kudos:
895
 [2]
Given Kudos: 40
GPA: 3.98
Posts: 297
Kudos: 895
 [2]
How many integers less than 100 have exactly 4 odd factors b 31 Oct 2016, 03:43
1
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
In order to achieve requested 4 factors the number should be in the form p^3 or p*q (giving 3+1 and 2+2 total number of factors), where p, and q are odd prime factors.
In first case we have only one match - 3^3 = 27
Second case we’ll calculate in ascending order to evade double counting:
3*5=15
3*7=21
3*11=33
….
3*31=93
____________
5*7=35
5*11=55
…
5*19=95
____________
7*11=77
7*13=91
Total 16

Total number of integers = 16+1=17

Answer D.
User avatar
Shazzad777
User avatar
Joined: 16 Jul 2023
Last visit: 12 Sep 2023
Posts: 2
Own Kudos:
6
Given Kudos: 1
Status:Educator
Posts: 2
Kudos: 6
How many integers less than 100 have exactly 4 odd factors b 13 Aug 2023, 05:32
Kudos
Add Kudos
Bookmarks
Bookmark this Post
vitaliyGMAT
In order to achieve requested 4 factors the number should be in the form p^3 or p*q (giving 3+1 and 2+2 total number of factors), where p, and q are odd prime factors.
In first case we have only one match - 3^3 = 27
Second case we’ll calculate in ascending order to evade double counting:
3*5=15
3*7=21
3*11=33
….
3*31=93
____________
5*7=35
5*11=55
…
5*19=95
____________
7*11=77
7*13=91
Total 16

Total number of integers = 16+1=17

Answer D.
Show more


Why didn't you include 5, 9 = 45 and 7, 9 = 63? That makes sense right?
Moderators:
Bunuel
Math Expert
109785 posts
Krunaal
Tuck School Moderator
853 posts