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Factorial is an important topic in quantitative aptitude preparation not just because of its importance as an independent concept, but also because of its application in many topics like permutation & Combination, Probability etc… The factorial of a non negative integer n is denoted as n! The notation was introduced by Christian Kramp in 1808.

n! is calculated as the product of all positive integers less than or equal to n.

i.e 6! = 1 * 2 * 3 * 4 * 5 * 6 = 720

n! = 1 when n = 0, and n! = (n - 1)! * n if n > 0

**➢ What is the significance of n! ****"n! **is the number of ways we can arrange n distinct objects into a sequence**".****Example: **2! = 2 means numbers 1, 2 can be arranged in 2 sequences (1, 2) and (2, 1).

We can arrange 0 in one way. So 0! = 1, not zero. Now we know why, and no need to say “its like that” if someone asks.

➢** Find the highest power of a prime number in a given factorial.**

The highest power of prime number p in **n! = [n/p ^{1}] + [n/p^{2}] + [n/p^{3}] + [n/p^{4}] + …**

[x] denotes the greatest integer less than or equal to x.

[1.2] =1

[4] = 4

[-3.4] = -4 (why? Because -4 is less than -3.2)

**Example 1: Find the highest power of 3 in 100!****Solution:** [100/3] + [100/3^{2}] + [100/3^{3}] + [100/3^{4}] + [100/3^{5}] + …

= 33 + 11 + 3 + 1 + 0 (from here on greatest integer function evaluates to zero)

= 48.

**Example 2: **What is the greatest power of 5 which can divide 80! exactly?**Solution:** [80/5] + [80/5^{2}] + [80/5^{3}] + …

= 16 + 3 + 0 + … = 19.

**➢ Find the highest power of a non prime number in a given factorial**

We learnt before that any non prime number can be expressed as a product of its prime factors. We will use this concept to solve this type of questions.

**Example 1: What is the greatest power of 30 in 50!****Solution: **30 = 2 * 3 * 5.

Find what the highest power of each prime is in the given factorial

We have 25 + 12 + 6 + 3 + 1 = 47 2s in 50!

16 + 5 + 1 = 22 3s in 50!

10 + 2 = 12 5s in 50!

We need a combination of one 2, one 3 and one 5 to get 30. In the given factorial we have only 12 5s. Hence only twelve 30s are possible. So the greatest power of 30 in 50! is 12.

Note:We don’t have to find the power of all prime factors of a given number. It is enough to find the greatest power of the highest prime factor in the factorial (here 5).

**Example 2: What is the greatest power of 8 in 50!****Solution: **8 =2^{3}

50! has 47 twos. We need 3 two to get an 8.

We can have a maximum of [47/3] = 15 8s in 50!

Note:The highest power of the number p^{a}in n! = [Highest power of p in n!]/a.

**➢ **Find the highest power of all prime numbers, less than n, in n!**Example 1: What are the number of factors of 15!****Solution: **2, 3, 5, 7, 11 and 13 are the prime numbers less than 15.

Highest power of 2 in 15! = 11

Highest power of 3 in 15! = 6

Highest power of 5 in 15! = 3

Highest power of 7 in 15! = 2

Highest power of 11 in 15! = 1

Highest power of 13 in 15! = 1

15! = 2^{11 }* 3^{6 }* 5^{3 }* 7^{2}* 11^{1 }* 13^{1}

Thus we can find the number of divisors, sum of divisors and other values for 15!

number of factors of 15! = (11 +1) (6 + 1) (3 + 1) ( 2 + 1) ( 1 + 1) (1 + 1) = 4032

**➢ **Find the number of zeros at the end of a factorial value

This is just another way of asking what highest power of 10 is in a given factorial.

10 = 2*5, we need a combination of one 2 and one 5 to get a zero at the end. Just find the number of 5s in the given factorial as number of 2s will be always more than number of 5s.

**Example 1: Find the number of zeros at the end of 100!****Solution:** [100/5] + [100/25] + … = 20 + 4 + 0 + … = 24.

**Example 2: **Find the number of zeros at at the end of 100! in base 7**Solution:** Concept is same as before. in base 10 we need a 2 and 5 to get a zero but in base 7 we need a 7 to get a zero. so we have to find the highest power of 7 in 100!

[100/7] + [100/49] + ... = 14 + 2 = 16. There will be 16 zeros at the end of 100! in base 7

Note:To find the number of zeros of a factorial in a base b, calculate the highest power of b in that factorial. b need not be prime, we already know how to find the highest power of a composite number in a factorial expression.

**WILSON'S THEOREM****For every prime number p, (p-1)! + 1 is divisible by p**

Take p =7, then as per Wilson's theorem 7 is prime only if (6!) + 1 is divisible by 7 and some useful results derived from Wilson's theorem.

Remainder [(p-1)!/p] = p - 1 ( where p is a prime number )

Remainder [6!/7] = 7; Remainder [100!/101] = 100 and so on...

Remainder [(p-2)!/p] = 1 ( where p is a prime number )

Remainder [5!/7] = 1; Remainder [99!/101] = 1 and so on...

**SOME EXTRA CONCEPTS**** ➢ **Product of n consecutive natural numbers is divisible by n!

11 * 12 * 13 * 14 is completely divisible by 4!

This can be extended to obtain other interesting results.

Consider 1 * 2 * 3 * 4 * 5 * 6 . as per above result 1 * 2 * 3 * 4 * 5 * 6 is divisible by 6!.

Now take (1 * 2 * 3) * (4 * 5 * 6) . each of this is divisible by 3!

so 1 * 2 * 3 * 4 * 5 * 6 is completely divisible by (3!)

similarly 1 * 2 * 3 * 4 * 5 * 6 is completely divisible by (2!)

Note:To generalise (a*n)! is completely divisible by (n!)^{a}, where a and n are integers.

** ➢ **The product of all odd (even) integers up to an odd (even) positive integer n is called the double factorial of n. (though it has only about half the factors of the original factorial!). It is denoted as n!!

9!! = 1 * 3 * 5 * 7 * 9 = 945

8!! = 2 * 4 * 6 * 8 = 384

** ➢ **Product of first n! is called the super factorial of n, denoted as sf(n).

sf(4) = 1! * 2! * 3! * 4!

** ➢ **A factorion is a natural number that equals the sum of the factorials of its decimal digits.

**Note: **There are just four factorions (in base 10) and they are 1, 2, 145 and 40585.

** ➢ **A factorial prime is a prime number that is one less or one more than a factorial.

2 = 1! + 1, 3 = 2! + 1, 5 = 3! -1, 7 = 3! + 1 etc…

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