Last visit was: 23 Apr 2026, 04:50

It is currently 23 Apr 2026, 04:50

Toolkit

Practice Tests

Quantitative Questions

Problem Solving (PS)

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

Request Expert Reply

Confirm Cancel

Apr 26
Score High on the GMAT with Target Test Prep
10:00 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 27
Try OnDemand GMAT Masterclass
11:00 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 28
Join - GMAT Day!
08:00 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.

Back to Forum

Create Topic

GMAT Club Olympics 2025 (Day 3): If a positive integer n has 24

1 2 3 4 5 6

Bunuel

Math Expert

Joined: 02 Sep 2009

Last visit: 23 Apr 2026

Posts: 109,776

Own Kudos:

810,754

[30]

Given Kudos: 105,853

Products:

Expert

Expert reply

Posts: 109,776

Kudos: 810,754

[30]

02 Jul 2025, 08:00

Kudos

Bookmarks

00:00

Start Timer

Pause Timer

Resume Timer

Show Answer

Hide

Show

History

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:

44% (01:42) correct

56% (01:44) wrong

based on 555 sessions

History

Date

Time

Result

Not Attempted Yet

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

Show Answer

Official Answer

Most Helpful Reply

Bunuel

Math Expert

Joined: 02 Sep 2009

Last visit: 23 Apr 2026

Posts: 109,776

Own Kudos:

810,754

[4]

Given Kudos: 105,853

Products:

Expert

Expert reply

Posts: 109,776

Kudos: 810,754

[4]

02 Jul 2025, 08:30

Kudos

Bookmarks

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

GMAT Club Official Explanation:

If the prime factorization of a positive integer is a^x * b^y * c^z *... where a, b, and c are prime numbers and x, y, and z are their powers, then the number of positive factors of the integer is given by (x + 1)(y + 1)(z + 1)... So, to find the number of positive factors, we add 1 to the powers of the distinct primes in the prime factorization and multiply.

We are given that n has 24 factors. To find how many prime factors n can have, we consider how 24 can be written as a product of such terms.

The least number of primes n can have is 1, if n = prime^23, which gives the number of factors as 23 + 1 = 24. For example, n could be 2^23.

For the maximum number of primes, break 24 into the maximum number of integers greater than 1: 24 = 2 * 2 * 2 * 3. In this case, n would have 4 primes: n = (prime_1) * (prime_2) * (prime_3) * (prime_4)^2, which gives the number of factors as (1 + 1)(1 + 1)(1 + 1)(2 + 1) = 2 * 2 * 2 * 3 = 24. For example, n could be 2 * 3 * 5 * 7^2.

So, the range is 4 - 1 = 3.

Answer: C.

General Discussion

APram

Joined: 23 Jun 2024

Last visit: 17 Mar 2026

Posts: 709

Own Kudos:

280

Given Kudos: 248

Location: India

Schools: ISB '26 HEC INSEAD Jan '26

GMAT Focus 1: 605 Q86 V78 DI76 Verified score

GPA: 3.608

Schools: ISB '26 HEC INSEAD Jan '26

GMAT Focus 1: 605 Q86 V78 DI76 Verified score

Posts: 709

Kudos: 280

02 Jul 2025, 08:12

Kudos

Bookmarks

Number of positive factors = n! where n is the number of distinct prime factors

24 = 4!
hence there can be maximum 4 distinct prime factors

Hence D is correct

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

Archit3110

Major Poster

Joined: 18 Aug 2017

Last visit: 23 Apr 2026

Posts: 8,628

Own Kudos:

5,190

[1]

Given Kudos: 243

Status:You learn more from failure than from success.

Location: India

Concentration: Sustainability, Marketing

GMAT Focus 1: 545 Q79 V79 DI73 Verified score

GMAT Focus 2: 645 Q83 V82 DI81 Verified score

GPA: 4

WE:Marketing (Energy)

Products:

GMAT Focus 2: 645 Q83 V82 DI81 Verified score

Posts: 8,628

Kudos: 5,190

[1]

02 Jul 2025, 08:14

Kudos

Bookmarks

total positive factors 24
possible when power of numbers is 2*2*2*3
i.e. number has to be prime to have 2 has power ; so total prime numbers will be 3
OPTION C, 3

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

simondahlfors

Joined: 24 Jun 2025

Last visit: 23 Sep 2025

Posts: 48

Own Kudos:

[7]

Posts: 48

Kudos: 46

[7]

02 Jul 2025, 08:14

Kudos

Bookmarks

Information given:
- A positive integer n has 24 positive factors

Question:
- What is the maximum possible range in the number of distinct prime factors that n can have

Solution:
- The number of factors formula for n = p1^a * p2^b * p3^c ... is factors = (a + 1)(b + 1)(c + 1)
- To maximize the number of distinct primes, split 24 into as many integer factors greater than or equal to 2 as possible
- 24 = 2 x 2 x 2 x 3 = 4 factors (and 4 primes)
- Therefore, the maximum possible distinct prime factors is 4
- The minimum number of distinct prime factors occurs when you use only one prime raised to a power (e.g. 2^23), which would give 1 distinct prime factor
- The maximum possible range therefore is 4 - 1 = 3

Answer: C, 3

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

D3N0

Joined: 21 Jan 2015

Last visit: 19 Mar 2026

Posts: 585

Own Kudos:

607

[1]

Given Kudos: 132

Location: India

Concentration: Operations, Technology

Schools: Smeal'21 (WD) Arizona State'21 (A$) Simon '21 (WD)

GMAT 1: 620 Q48 V28 Verified score

GMAT 2: 690 Q49 V35 Verified score

WE:Operations (Retail: E-commerce)

Products:

Schools: Smeal'21 (WD) Arizona State'21 (A$) Simon '21 (WD)

GMAT 2: 690 Q49 V35 Verified score

Posts: 585

Kudos: 607

[1]

02 Jul 2025, 08:16

Kudos

Bookmarks

Ans: C (3)

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

number of positive factors = prime^a * prime^b * prime^c = (a+1)* (b+1) * (c+1) * (d+1)

n has 24 positive factors = 24 = 3*2*2*2 = (a+1)* (b+1) * (c+1) * (d+1)
minimum (some value)^23 will give us (23+1 = 24 factors) so one prime value

max when all 4 ( (a+1)* (b+1) * (c+1) * (d+1) ) are on different prime factors
in this case 4 distinct prime factors

range = max - min
4-1 = 3

MaxFabianKirchner

Joined: 02 Jun 2025

Last visit: 12 Jul 2025

Posts: 65

Own Kudos:

[2]

Given Kudos: 122

Status:26' Applicant

Location: Denmark

Concentration: Finance, International Business

Schools: ESSEC MiF '26 ESADE HEC Imperial LBS Oxford MFE '26 Sloan (MIT) NYU Stern NUS Bocconi IE'26

GPA: 4.0

WE:Other (Student)

Schools: ESSEC MiF '26 ESADE HEC Imperial LBS Oxford MFE '26 Sloan (MIT) NYU Stern NUS Bocconi IE'26

Posts: 65

Kudos: 63

[2]

02 Jul 2025, 08:18

Kudos

Bookmarks

The number of factors of an integer is found by taking the exponents in its prime factorization, adding 1 to each, and multiplying the results. We are given that this product is 24. The number of distinct prime factors is the number of terms in this product.

To find the minimum number of distinct prime factors, we use the fewest possible terms to make 24. We can use one term: 24.

This corresponds to a number like p^23, which has only 1 distinct prime factor.

To find the maximum number of distinct prime factors, we use the most possible terms. We break 24 into its smallest factors (greater than 1): 24 = 2 * 2 * 2 * 3.

This product has four terms, corresponding to a number like p1^1 * p2^1 * p3^1 * p4^2, which has 4 distinct prime factors.

The number of distinct prime factors can range from a minimum of 1 to a maximum of 4.

The range is the difference: 4 - 1 = 3.

The correct answer is (C) 3.

Kinshook

Major Poster

Joined: 03 Jun 2019

Last visit: 23 Apr 2026

Posts: 5,986

Own Kudos:

5,858

[1]

Given Kudos: 163

Location: India

Schools: HBS '26 CBS '26 Wharton '26

GMAT 1: 690 Q50 V34 Verified score

WE:Engineering (Transportation)

Products:

Schools: HBS '26 CBS '26 Wharton '26

GMAT 1: 690 Q50 V34 Verified score

Posts: 5,986

Kudos: 5,858

[1]

02 Jul 2025, 08:19

Kudos

Bookmarks

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

For minimum number of prime factors, n will be of the form p^23
For maximum number of prime factor, n will be of the form p1*p2*p3*p4^2

Maximum possible range of the number of distinct prime factors that n can have = 4 - 1 = 3

IMO C

Heix

Joined: 21 Feb 2024

Last visit: 13 Apr 2026

Posts: 460

Own Kudos:

165

[1]

Given Kudos: 63

Location: India

Concentration: Finance, Entrepreneurship

Schools: Ross MM "26 NUS Columbia '27

GMAT Focus 1: 485 Q76 V74 DI77 Verified score

GPA: 3.4

WE:Accounting (Finance)

Products:

Schools: Ross MM "26 NUS Columbia '27

GMAT Focus 1: 485 Q76 V74 DI77 Verified score

Posts: 460

Kudos: 165

[1]

02 Jul 2025, 08:19

Kudos

Bookmarks

First, I'll recall how to determine the number of factors of a number. If a number n has prime factorization: n=paxpox pcx... *pk
Then the total number of factors of n is:
(a+1) * (b+1) × (c+1)x ... * (k+1)
Since we know the number has exactly 24 factors, I need to find the prime factorization combinations that give us 24 factors.
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
So the possible combinations for (a+1) × (b+1) x ... that equal 24 are:
* 24 = 24 × 1 → This means one prime raised to power 23
* 24 = 12 × 2 → This means one prime raised to power 11 and another to power 1
* 24 = x x 3→ This means one prime raised to power 7 and another to power 2
* 24=6 x 4→ This means one prime raised to power 5 and another to power 3
* 24=6×2 x 2→ This means one prime raised to power 5 and two others each to power 1
* 24=4x3 × 2 → This means one prime raised to power 3, another to power 2, and a third to power 1
* 24=3x2x2x2→ This means one prime raised to power 2 and three others each to power 1
* 24=2x2x2x3*1→ This means four primes each raised to power 1 and one prime raised to power 2
The last case I listed isn't valid because it would require 5 distinct primes but the product of exponents would be 2x2x2x3x1 = 24, not 24 factors.
From these combinations:
* Minimum number of distinct prime factors: 1 (from the first case)
* Maximum number of distinct prime factors: 4 (from the second-to-last case)
Therefore, the range in the number of distinct prime factors is 4 - 1 = 3.
The answer is C. 3.

Emkicheru

Joined: 12 Sep 2023

Last visit: 12 Sep 2025

Posts: 119

Own Kudos:

Given Kudos: 11

Location: Kenya

Schools: Anderson (UCLA) Sloan (MIT) Columbia '27

GMAT 1: 780 Q50 V48 Verified score

GRE 1: Q167 V164 Verified score

GPA: 3.7

Schools: Anderson (UCLA) Sloan (MIT) Columbia '27

GMAT 1: 780 Q50 V48 Verified score

GRE 1: Q167 V164 Verified score

Posts: 119

Kudos: 22

02 Jul 2025, 08:21

Kudos

Bookmarks

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

Taking n as 24 ,we know there are 8 prime numbers in 24 therefore from the choices the larger number is 5 which is less than 8, hence we take 5 as the maximum possible prime

iamchinu97

Joined: 14 Dec 2020

Last visit: 19 Apr 2026

Posts: 136

Own Kudos:

145

[1]

Given Kudos: 35

Products:

Posts: 136

Kudos: 145

[1]

02 Jul 2025, 08:26

Kudos

Bookmarks

Ok for min prime factor we know we can have p^23 so 1
for max prime factors lets break down 24 = 3 * 2* 2* 2 => here p1^2 * p2 * p3 * p4 so 4
Range = 4-1 = 3
hence Ans C

HarshaBujji

Joined: 29 Jun 2020

Last visit: 05 Apr 2026

Posts: 723

Own Kudos:

906

[1]

Given Kudos: 247

Location: India

Products:

Posts: 723

Kudos: 906

[1]

02 Jul 2025, 08:27

Kudos

Bookmarks

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

Wow great onw, Almost got me wrong.

Always clearly define what we are looking for : Range = (Max #Prime factors - Min #Prime factors).

So there is a number that has 24 factors we need to find max,min prime factors for it to get the solution.

We know that #factors of N = $p1^(x1) + p2^(x2)+p3^(x3)................$ is (x1+1)(x2+1)......

So here to get min prime factors the number can be$ p^23 $ . So factors are 24. (1,p, p^2,p^3.....p^23)

To get the max prime factors we need have as more x1,x2,..
So 24 can be written as 2*2*2*3 => So $ (p1) * (p2) * (p3) * (p4)^2$ can be the number, Max prime factors is 4.

Hence, the range is 4-1 =3, IMO C

Rahul_Sharma23

Joined: 05 Aug 2023

Last visit: 22 Apr 2026

Posts: 116

Own Kudos:

Given Kudos: 17

Location: India

Schools: Haas '27 Fuqua '27 Darden '27 ISB '27 (S) McCombs '27 HEC Yale '27 Rotman '27

GMAT Focus 1: 695 Q87 V83 DI83 Verified score

GPA: 2.5

Products:

Schools: Haas '27 Fuqua '27 Darden '27 ISB '27 (S) McCombs '27 HEC Yale '27 Rotman '27

GMAT Focus 1: 695 Q87 V83 DI83 Verified score

Posts: 116

Kudos: 83

02 Jul 2025, 08:27

Kudos

Bookmarks

number of factors = (p+1)*(q+1)*...

where p, q are power of prime numbers in factorization

24 = 2*2*2*3
24 = (p+1)(q+1)(r+1)(s+1) , where p=1, q=1, r=1, s=2

therefore max 4 distinct prime factors are possible

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

Cana1766

Joined: 26 May 2024

Last visit: 22 Apr 2026

Posts: 85

Own Kudos:

[1]

Given Kudos: 12

Posts: 85

Kudos: 80

[1]

02 Jul 2025, 08:31

Kudos

Bookmarks

Here the integer has 24 factors.
We can write the integer as distinct prime factors
Like i=p1^a1*p2^a2*p3^a3........(here p1,p2 are prime numbers like 2,3 etc)
But to get the total factors from this equation we have to do (a1+1)(a2+1)(a3+1).....=24(given from question)

Next we have to figure out a1,a2

They have asked range.Lets think of maximum.If a1=23,it satisfies the equation. That means i=p1^23
What about minimum ,we can break 24 as 2*2*2*3,So that means a1=1,a2=1,a3=1,a4=2,that means i=p1^1*p2^1,p3^1,p4^2

Finally, we can now see that we have 1 prime number in the first case and 4 prime numbers in the second case.
Range is 4-1=3
Answer is C.

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

BeachStudy

Joined: 30 Jun 2025

Last visit: 18 Aug 2025

Posts: 61

Own Kudos:

[1]

Given Kudos: 4

Posts: 61

Kudos: 37

[1]

02 Jul 2025, 08:39

Kudos

Bookmarks

Integer: a whole number (not a fraction or decimal) that can be positive, negative, or zero.
Factor: number that divides another number evenly, resulting in a whole number with no remainder
Distinct prime factors: the unique prime numbers that divide that number.

When it comes to prime factors we can do 3*2*2*2
Lowest range is 1. Highest is 4.
So C - 3

Answer C

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

bart08241192

Joined: 03 Dec 2024

Last visit: 28 Dec 2025

Posts: 75

Own Kudos:

[1]

Given Kudos: 13

Posts: 75

Kudos: 64

[1]

02 Jul 2025, 08:43

Kudos

Bookmarks

if we want to calculate the the number of factors

n=2^(a+1) 3(b+1).....

if we put all 24 factor in one prime
a+1=24 ,the prime number is only 1
if we separate 24 factors into different prime
each one just 1
just like
2*3*5*7....
(a+1)(b+1)(c+1)...

2x2x2x3=24
that means we can have 4 prime numbers
range=4-1=3

dthaman1201

Joined: 08 Dec 2024

Last visit: 11 Mar 2026

Posts: 23

Own Kudos:

[1]

Given Kudos: 165

Posts: 23

Kudos: 18

[1]

02 Jul 2025, 08:48

Kudos

Bookmarks

For any number who has prime no in factors their total factors are +1 to their exponent and then all powers are multiplied together.

We know total factors are 24
so
24 = (23+1) ---> 1 factor only
24= 2 * 12---> (1+1)(11+1) --> 2 factor
24= 2*6*2-->(1+1)(5+1)(1+1)-->3 factor
24=2*2*3*2--->4 factor

range = 3

Aashimabhatia

Joined: 29 Aug 2024

Last visit: 02 Apr 2026

Posts: 45

Own Kudos:

[1]

Given Kudos: 656

Posts: 45

Kudos: 26

[1]

02 Jul 2025, 08:52

Kudos

Bookmarks

Total Number of Factors = Product of (Powers of distinct prime factors + 1)

24= 2*2*2*3
This means, maximum no of Distinct Prime factors of n is 4
(n= a^1* b^1* c^1* d^2 where a,b,c,d are distinct prime factors)

Minimum no of distinct Prime factors of n is 1
(n= e^23)

Range= Max-Min= 4-1= 3

Ans- 3

Duongduong273

Joined: 08 Apr 2023

Last visit: 03 Feb 2026

Posts: 32

Own Kudos:

[1]

Given Kudos: 4

Posts: 32

Kudos: 32

[1]

02 Jul 2025, 08:52

Kudos

Bookmarks

Number of factors: (a+1)(b+1)(c+1)...=24. Maximum possible range in the number of distinct prime factors when break 24 into many 2 as possible: 24=2^3x3=2x2x2x3, so 4 different prime factors is maximum. Minimum is 1 prime factor when n = a prime ^ 23, n will has 24 factors: (23+1)=24=> Range: max-min: 4-1=3 => C

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

bhanu29

Joined: 02 Oct 2024

Last visit: 22 Apr 2026

Posts: 358

Own Kudos:

270

[1]

Given Kudos: 263

Location: India

Schools: Anderson '27 Booth '27 Foster '27 Haas '27 Kellogg '27 Kellogg Mashall '27 McCombs '27 Ross '27 Scheller '27 Tepper '27 Yale '27 Sloan (MIT) Stanford '27

GMAT Focus 1: 675 Q87 V85 DI79 Verified score

GMAT Focus 2: 715 Q87 V84 DI86 Verified score

GPA: 9.11

WE:Engineering (Technology)

Products:

Schools: Anderson '27 Booth '27 Foster '27 Haas '27 Kellogg '27 Kellogg Mashall '27 McCombs '27 Ross '27 Scheller '27 Tepper '27 Yale '27 Sloan (MIT) Stanford '27

GMAT Focus 2: 715 Q87 V84 DI86 Verified score

Posts: 358

Kudos: 270

[1]

02 Jul 2025, 08:56

Kudos

Bookmarks

Bunuel

If a positive integer n has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that n can have?

A. 1
B. 2
C. 3
D. 4
E. 5

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

If a number X can be expressed as, X= p1^a1 * p2^a2 ... pn^an

where p1, p2, ... pn are distinct prime factors

Then total factors = (a1+1)*(a2+1)*....(an+1)

24 can be expressed as

24 = (23+1), n could be p^23, distinct prime factor = 1

It can also be expressed as 24 = 2 * 2 * 2 * 3 = (1+1) * (1+1) * (1+1) * (2+1) , n could be p1^1 *p2^1 * p3^1 *p4^2, distinct prime factors =4

Maximum Range = 4-1 = 3

Answer choice B is the correct answer