Last visit was: 17 Jun 2025, 23:21 It is currently 17 Jun 2025, 23:21
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.
Close
Request Expert Reply
Confirm Cancel
User avatar
EgmatQuantExpert
User avatar
e-GMAT Representative
Joined: 04 Jan 2015
Last visit: 02 Apr 2024
Posts: 3,681
Own Kudos:
19,314
 [34]
Given Kudos: 165
Expert
Expert reply
Posts: 3,681
Kudos: 19,314
 [34]
5
Kudos
Add Kudos
29
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
EgmatQuantExpert
User avatar
e-GMAT Representative
Joined: 04 Jan 2015
Last visit: 02 Apr 2024
Posts: 3,681
Own Kudos:
19,314
 [7]
Given Kudos: 165
Expert
Expert reply
Posts: 3,681
Kudos: 19,314
 [7]
2
Kudos
Add Kudos
5
Bookmarks
Bookmark this Post
General Discussion
User avatar
Hatakekakashi
Joined: 07 Jan 2016
Last visit: 22 Feb 2025
Posts: 1,245
Own Kudos:
477
 [3]
Given Kudos: 126
Location: United States (MO)
GMAT 1: 710 Q49 V36
Products:
GMAT 1: 710 Q49 V36
Posts: 1,245
Kudos: 477
 [3]
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
User avatar
EgmatQuantExpert
User avatar
e-GMAT Representative
Joined: 04 Jan 2015
Last visit: 02 Apr 2024
Posts: 3,681
Own Kudos:
19,314
 [2]
Given Kudos: 165
Expert
Expert reply
Posts: 3,681
Kudos: 19,314
 [2]
1
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post

Solution:



We are given:
\(f(n)= n!\)
Thus we can write:
    \(f(10)= 10!\)
    \(10! = 1×2×3×4×5×6×7×8×9×10\)
    \(10!= 2^8 * 3^4 * 5^2 *7\)
Total factors of \(10!= 9*5*3*2=270\)
Total factors = Even factors + Odd factors
Odd factors= Total factors – Even factors
Thus, if we can find the even factors of \(10!\) then we can find the odd factors of \(10!\).
\(10!= 2*(2^7 * 3^4 * 5^2 *7)\)
We can say that all the factors of \(2^7 * 3^4 * 5^2 *7\) after multiplying by \(2\) becomes a factor of \(10!\).
We also know that any number, which is multiple of \(2\), is an even number.
Thus,
    Total number of factors of\(2^7 * 3^4 * 5^2 *7\) = Even number of factors of \(10!.\)
    Even factors of \(10!\)= Total factors of \(2^7 * 3^4 * 5^2 *7\)
    Even factors of \(10!\)= \(8*5*3*2\)
    Even factors of \(10!\)=\(240\)
Therefore, Odd factors= Total factors – Even factors
Odd factors= \(270-240\)
Odd factors=\(30\)
Answer: Option \(A\)
User avatar
MikeScarn
User avatar
Current Student
Joined: 04 Sep 2017
Last visit: 01 Jun 2025
Posts: 278
Own Kudos:
1,249
 [4]
Given Kudos: 228
Location: United States (IL)
Concentration: Technology, Leadership
GMAT 1: 690 Q44 V41
GMAT 2: 730 Q50 V38
GPA: 3.62
WE:Sales (Computer Software)
2
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
EgmatQuantExpert

e-GMAT Question:



If \(f(n)\) is the product of \(n\) consecutive integers from \(1\) to \(N\), then what is the number of odd factors of\(f(10)\).
A) 30
B) 60
C) 120
D) 256
E) 512

This is

Question 11 of The e-GMAT Number Properties Marathon




Go to

the next level of the Marathon


The principle this question is testing is Finding the Number of Factors of an Integer

Number of odd factors of 10!

10!=1x2x3x4x5x6x7x8x9x10.
10!=2x3x(\(2^{2}\))x5x(2x3)x7x(\(2^{3}\))x(\(3^{2}\))x(5x2)
10!=\(2^{8}\)\(3^{4}\)\(5^{2}\)7

Take the exponents of the odd factors. Add 1 to each of them and multiply them by each other.
(4+1)(2+1)(1+1)=30

Answer = A
Attachments

Prime factors.PNG
Prime factors.PNG [ 29.38 KiB | Viewed 8546 times ]

User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 14 Jun 2025
Posts: 1,928
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,928
Kudos: 6,358
Kudos
Add Kudos
Bookmarks
Bookmark this Post
HI
I feel question would have been more interesting if there is an option 270 in the answer choices.
If someone fail to read odd factors (as I did ;) ), one would mark 270 as answer.
Fortunately, answer choices saved me.

Regards

EgmatQuantExpert

Solution:



We are given:
\(F(n)= 1×2×3×4×5×……….×n\)
We need to find the number of odd factors of \(f(10)\).
Thus, we need to find the value of \(f(10).\) Then, we need to write \(f(10)\) in its prime factorized form. After which we can calculate the value of odd factors of\(f(10)\).
Number of Odd factors \(f(10)\)
    \(f(10)= 1×2×3×4×5×6×7×8×9×10\)
    \(f(10)= 1×2×3×(2×2)×5×(2*3)×7×(2*2*2)×(3*3)×(2*5)\)
    \(f(10)= 2^8 * 3^4 * 5^2 *7\)
Odd factors of \(f(10)\)= (power of 3+1)*(power of 5+1)* (power of 7+1)
Odd factors of \(f(10)=5*3*2\)
Odd factors of \(f(10)=30\)
avatar
kabirchaudhry92
Joined: 27 May 2018
Last visit: 19 Aug 2018
Posts: 6
Own Kudos:
4
 [1]
Given Kudos: 28
Posts: 6
Kudos: 4
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
EgmatQuantExpert

Solution:



We are given:
\(F(n)= 1×2×3×4×5×……….×n\)
We need to find the number of odd factors of \(f(10)\).
Thus, we need to find the value of \(f(10).\) Then, we need to write \(f(10)\) in its prime factorized form. After which we can calculate the value of odd factors of\(f(10)\).
Number of Odd factors \(f(10)\)
    \(f(10)= 1×2×3×4×5×6×7×8×9×10\)
    \(f(10)= 1×2×3×(2×2)×5×(2*3)×7×(2*2*2)×(3*3)×(2*5)\)
    \(f(10)= 2^8 * 3^4 * 5^2 *7\)
Odd factors of \(f(10)\)= (power of 3+1)*(power of 5+1)* (power of 7+1)
Odd factors of \(f(10)=5*3*2\)
Odd factors of \(f(10)=30\)

Hello!

I have a doubt here.. Why is 1 not included in the odd factors ?
Sorry for such a silly question but isn't that supposed to be counted too ?

It would be really helpful if you could provide your insights on the same.

Thanks in advance
User avatar
ganand
Joined: 17 May 2015
Last visit: 19 Mar 2022
Posts: 198
Own Kudos:
3,517
 [1]
Given Kudos: 85
Posts: 198
Kudos: 3,517
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
kabirchaudhry92
EgmatQuantExpert

Solution:



We are given:
\(F(n)= 1×2×3×4×5×……….×n\)
We need to find the number of odd factors of \(f(10)\).
Thus, we need to find the value of \(f(10).\) Then, we need to write \(f(10)\) in its prime factorized form. After which we can calculate the value of odd factors of\(f(10)\).
Number of Odd factors \(f(10)\)
    \(f(10)= 1×2×3×4×5×6×7×8×9×10\)
    \(f(10)= 1×2×3×(2×2)×5×(2*3)×7×(2*2*2)×(3*3)×(2*5)\)
    \(f(10)= 2^8 * 3^4 * 5^2 *7\)
Odd factors of \(f(10)\)= (power of 3+1)*(power of 5+1)* (power of 7+1)
Odd factors of \(f(10)=5*3*2\)
Odd factors of \(f(10)=30\)

Hello!

I have a doubt here.. Why is 1 not included in the odd factors ?
Sorry for such a silly question but isn't that supposed to be counted too ?

It would be really helpful if you could provide your insights on the same.

Thanks in advance

Hi kabirchaudhry92,

This is a good question. Let me explain with some small examples.

Q1. Find the number of factors of 25.

Step 1: Do the prime factorization. 25 = 5*5 = \(5^2\)

Step 2: Number of factors = (2+1) = 3.

Why have we added 1? By adding "1" we ensure that we count 1 as one of the factors.

Factors of 25: \(5^0 = 1, 5^1 = 5,\) and \(5^2 = 25.\) Hence, the answer is 3.

Q2. Find the number of factors of 75.

Step 1: Prime factorization = 3*5*5 = \(3^1 * 5^2\)

Step 2: Number of factors = (1+1) * (2+1) = 6.

Factors of 75:
  • \(5^0 * 3^0 = 1\)
  • \(5^0 * 3^1 = 3\)
  • \(5^1 * 3^0 = 5\)
  • \(5^1 * 3^1 = 15\)
  • \(5^2 * 3^0 = 25\)
  • \(5^2 * 3^1 = 75\)
.

Hence, 1 is already included in the given solution.

I hope this helps.

Thank you.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 37,220
Own Kudos:
Posts: 37,220
Kudos: 1,000
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Moderators:
Math Expert
102085 posts
PS Forum Moderator
644 posts