CAT Factorial is an important topic for preparations. It is part of the number system questions. As number system contains 10% weightage in CAT exams. Aspirants should expect a few questions from Factorial. As quantitative section contain a total of 34 questions and out of that few questions are from number system problems. A factorial is the most formula and fact-based topic.

Clearing your factorial concepts will enhance your accuracy level in exams. Hence, Pay attention to concepts and formulas. As you all know number system topics contain 10% of the weightage in the CAT exam. Hence this become an important topic for exams.

In this article, we learn about some of the important dimensions of CAT factorial. All important aspects related to the topic will be covered. The aspirant would understand how to approach questions when his concepts become clear. In addition, one could easily ace factorial just by formulas and basic concepts clarity.

Also read: How to Prepare for CAT

The basic concept of Factorial for CAT exam.

As you all know factorials are considered from (n!). These factorial concepts look easy and simple but the problems could get complicated. So clear your concepts clearly for solving those complicated problems. Most aspirants preferred to solve factorial concepts over Permutation & Combination or Probability.

Exclamation marks denote factorial. Factorial is a multiplication of natural and real numbers. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6. In this article, you will learn the mathematical definition of the factorial, its notation, formula, examples and so on in detail.

In conclusion, Some of the factorials that might speed up your calculation are:

0! = 1; 1! is equal to 1; 2! = 2; 3! is equal to 6; 4! = 24; 5! is equal to 120; 6! = 720; 7! is equal to 5040.

Factorial Formula for CAT exam.

In sum the formula to find the factorial of a number is

n! = n × (n-1) × (n-2) × (n-3) × ….× 3 × 2 × 1

For an integer n ≥ 1, the factorial representation in terms of pi product notation is:

From the above formulas, the recurrence relation for the factorial of a number. Which is defined as the product of the factorial number. Factorial of that number minus 1. It is given by:

n! = n. (n-1)!

CAT Factorial of Numbers 1 to 10 Table

So Let’s have a look at the table below to understand the list of factorial values from 1 to 10 are:

CAT Factorial Rightmost non-zero digit of n! or R(n!)

R(n!) = Last Digit of [ 2^a x R(a!) x R(b!) ]

where n = 5a + b

For Example 1.1: What is the rightmost non-zero digit of 37!?

• R (37!) = Last Digit of [ 2⁷ x R (7!) x R (2!) ]
• R (37!) = Last Digit of [ 8 x 4 x 2 ] = 4

For Example 1.2: What is the rightmost non-zero digit of 134!?

• R (134!) = Last Digit of [ 2²⁶ x R (26!) x R (4!) ]
• R (134!) = Last Digit of [ 4 x R (26!) x 4 ]

We need to find out R (26!) = Last Digit of [ 2⁵ x R (5!) x R (1!) ] = the Last digit of [ 2 x 2 x 1 ] = 4

• R (134!) = Last Digit of [ 4 x 4 x 4 ] = 4

Power of a prime ‘p’ in a factorial (n!)

The biggest power of a prime ‘p’ that divides n! (or in other words, the power of prime ‘p’ in n!) is given by the sum of quotients obtained by successive division of ‘n’ by p.

For Example 2.1: What is the highest power of 7 that divides 1342?

• [1342 / 7] = 191
• [191 / 7] = 27
• [27 / 7] = 3
• Power of 7 = 191 + 27 + 3 = 221

For Example 2.2: What is the highest power of 6 that divides 134!?

As 6 is not a prime number, we will divide it into its prime factors. 3 is the bigger prime, so its power will be the limiting factor. Hence, we need to find out the power of 3 in 134!

• [134/3] = 44
• [44/3] = 14
• [14/3] = 4
• [4/3] = 1
• Power of 3 in 134! = 44 + 14 + 4 + 1 = 63

For Example 2.3: What is the highest power of 9 that divides 134!?

As 9 is not a prime number, we will divide it into its prime factors. 9 is 3². The number of 3s available is 63, so the number of 9s available will be [63/2] = 31.

The highest power of 9 that divides 134! is 31.

The highest power of 18 and 36 will also be 31. The highest power of 27 will be [63/3] = 21.

Notes to Remember

Note: To find out the highest power of a composite number, always try and find out which number (or prime number) will become the limiting factor. Use that to calculate your answer. In most cases, you can just look at a number and say which one of its prime factors will be the limiting factor. If it is not obvious, then you may need to find it out two of the prime factors. The above method can be used for doing the same.

CAT Factorial Number of ending zeroes in a factorial (n!)

The number of zeroes is given by the sum of the quotients obtained by successive division of ‘n’ by 5.

This is an extension of Factorial's Rightmost non-zero digit of n! or R(n!). The number of ending zeroes is nothing else but the number of times n! is divisible by 10 or in other words, the highest power of 10 that divides n!. 10 is not a prime number and its prime factors are 2 and 5. ‘5’ becomes the limiting factor and leads to the above-mentioned idea.

For Example 3.1: What is the number of ending zeroes in 134!?

• [134/5] = 26
• [26/5] = 5
• [5/5] = 1
• Several ending zeroes. Then, 26 + 5 + 1 = 32

Some sample questions with solutions.

For Example 1: What is the factorial of 6?

Solution: We know that the factorial formula is

n! = n × (n – 1) × (n – 2) × (n – 3) × ….× 3 × 2 × 1

So the factorial of 6 is

6! = 6 × (6 -1) × (6 – 2) × (6 – 3) × (6 – 4) × 1

6! = 6 × 5 × 4 × 3 × 2 ×1

Hence, 6! = 720

Therefore, the factorial of 6 is 720.

For Example 2: What is the factorial of 0?

Solution: The factorial of 0 is 1

i.e., 0 ! = 1

So According to the convention of empty product, the result of multiplying no factors is null. It means that the convention is equal to the multiplicative identity.

Some questions for practice factorial for CAT exam.

Question1. A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also, find the number of values of n that will satisfy these conditions.

Question2. Given N is a positive integer less than 31, how many values can n take if (n + 1) is a factor of n!?

Question3. How many values can natural number n take, if n! is a multiple of 7⁶ but not 7⁹?

Question4. Find the least number n such that no factorial has n trailing zeroes, or n + 1 trailing zeroes or n + 2 trailing zeroes.

Question5. Let K be the largest number with exactly 3 factors that divide 25! How many factors does (k – 1) have?

For CAT Quantitative section's full guidance visit this link- https://catking.in/best-guide-for-preparation-of-cat-quantitative-ability/

ENROL ON OUR COURSES NOW!

For CAT courses AND For Non-cat courses