CAT Quantitative Aptitude Questions | CAT Number Systems - Factorial questions

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

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.