CAT Quantitative Aptitude Questions | CAT Number Systems - Factorial questions

CAT Questions | Number Theory | Factorials - basic

The question is about basic factorial. Given a factorial of x, we need to find out the highest power of another number y that divides x! without leaving remainder. Factorials from CAT Number System is a fabulous idea and can be tested in multiple ways in the CAT Exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score in the CAT Exam. If you would like to take these CAT level questions as a Quiz,head on here to take these questions in a test format, absolutely free.

Question 6: What is the highest power of 12 that divides 54!?

  1. 25
  2. 26
  3. 30
  4. 4

Last day to avail 🎉 Flat ₹ 12,000 off on our CAT '24 & '25 Live Courses!


Register Now

🎉 2IIM CAT Revision Course 2024: 100+ Hours of Intensive Prep and everything is covered—Join Now!!! 🎉


Register Now

Best CAT Online Coaching
Try upto 40 hours for free
Learn from the best!


2IIM : Best Online CAT Coaching.


Best CAT Coaching in Chennai


CAT Coaching in Chennai - CAT 2022
Limited Seats Available - Register Now!


Explanatory Answer

Method of solving this CAT Question from Number Theory - Factorial: Finding highest power of a prime that divides any factorial is easy. How do we find the highest power of a composite number that divides a specific factorial.

12 = 22 * 3, so we need to count the highest power of 2 and highest power of 3 that will divide 54! and then we can use this to find the highest power of 12.
The method to find highest powers of 2 and 3 are similar to the one outlined in the previous question.
Highest power of 2 that divides 54! = [\\frac{54}{2}\\)] + [\\frac{54}{4}\\)] + [\\frac{54}{8}\\)] + [\\frac{54}{16}\\)] + [\\frac{54}{32}\\)]= 27 + 13 + 6 + 3 + 1 = 50
Highest power of 3 that divides 54! = [\\frac{54}{3}\\)] + [\\frac{54}{9}\\)] + [\\frac{54}{27}\\)] = 18 + 6 + 2 = 26

Or 54! is a multiple of 250 * 326. Importantly, these are the highest powers of 2 and 3 that divide 54!.
22 * 3 = 12. We need to see what is the highest power of 22 * 3 that we can accommodate within 54!
In other words, what is the highest n such that (22 * 3)n can be accommodated within 250 * 326.
Let us try some numbers, say, 10, 20, 30

(22 * 3)10 = 220 * 310, this is within 250 * 326
(22 * 3)20 = 240 * 320, this is within 250 * 326
(22 * 3)30 = 260 * 330, this is not within 250 * 326
The highest number possible for n is 25.
(22 * 3)25 = 250 * 325, this is within 250 * 326, but (22 * 3)26 = 252 * 326, this is not within 250 * 326.
So, 54! can be said to be a multiple of (22 * 3)25. Or, the highest power of 12 that can divide 54! is 25.
Note: For most numbers, we should be able to find the limiting prime. As in, to find the highest power of 10, we need to count 5s. For the highest power of 6, we count 3s. For 15, we count 5s. For 21, we count 7’s. However, for 12, the limiting prime could be 2 or 3, so we need to check both primes and then verify this.

The question is "What is the highest power of 12 that divides 54!?"

Hence the answer is "25"

Choice A is the correct answer.


CAT Online Coaching | CAT Number Systems questions Videos On YouTube


Other useful sources for Number System Questions | Number Theory Factorials Sample Questions


CAT Questions | CAT Quantitative Aptitude

CAT Questions | CAT DILR

CAT Questions | Verbal Ability for CAT


Where is 2IIM located?

2IIM Online CAT Coaching
A Fermat Education Initiative,
58/16, Indira Gandhi Street,
Kaveri Rangan Nagar, Saligramam, Chennai 600 093

How to reach 2IIM?

Phone: (91) 44 4505 8484
Mobile: (91) 99626 48484
WhatsApp: WhatsApp Now
Email: prep@2iim.com