Factorials :: Value, Addition, Subtraction, Multiplication, Division

What is a Factorial?

A factorial is a function whose domain is the set of whole numbers. It is defined as n!

n
∏
k=1

[n ≥ 0]

The factorial of a natural number "n" is the product of the all natural numbers less than or equal to "n"
[product (represented by ∏) of natural numbers starting from 1 and ending with the given number].

⇒ n! or ∠n = 1 x 2 x 3 x 4 x ... x n.

Representing Factorials

The symbol '!' [Exclamation] after the number or '∠' [Angle] before the number represents the factorial function.
[The "!" is also called "shriek", "bang" or "crit"]

Illustration
Factorial of 8 is the product of natural numbers starting with 1 and ending with 8.
⇒ 8! or ∠8 = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8

Factorial Interpretation

You will find factorials being used all throughout the topic permutations and combinations. However, it is a general idea which is used in many other topics in mathematics.

Expansion in Reverse Order
n! or ∠n = 1 × 2 × 3 × 4 × ... × n.
⇒ n! = 1 × 2 × 3 × 4 × ... × (n − 2) x (n − 1) × n.
⇒ n! = n × (n − 1) × (n − 2) × (n − 3) × ... × 3 × 2 × 1.
n! = n × (n − 1) × (n − 2) × (n − 3) × ... × 3 × 2 × 1.
⇒ n! = n × (n − 1)! [Since 1 x 2 x 3 x ... x (n - 1) = (n - 1)!]
⇒ n! = n × (n − 1) × (n − 2)! [Since 1 x 2 x 3 x ... x (n - 2) = (n - 2)!]
⇒ n! = n × (n − 1) × (n − 2) × (n − 3)! [Since 1 x 2 x 3 x ... x (n - 3) = (n - 3)!]
This would be most useful for simplifications of problems involving factorial notations.
Factorial of 1 (One)
n! = 1 x 2 x 3 x 4 x ...x (n − 2) x (n − 1) x n. ⇒ 1! = 1
Factorial of 0 (Zero)
0! = 1
Empty/Nullary Product
An empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is 1 (the multiplicative identity). [≡ Empty sum - Sum of no number is zero (the additive identity ) {0 + 0 = 0}]
Two most frequent instances of empty product are m⁰ = 1 (any number raised to the power zero is one) and 0! = 1 (the factorial of zero is one).

Arithmetic Operations involving Factorials

Multiplication

Product of two or more factorials is the product of their values.

Eg:

01.

5! × 12	=	(5 × 4 × 3 × 2 × 1) × 12
	=	120 × 12
	=	1,440

02.

5! × 6!	=	(5 × 4 × 3 × 2 × 1) x (6 × 5 × 4 × 3 × 2 × 1)
	=	120 x 720
	=	86,400

Division

The simplification process can be reduced by expanding the factorial notation.

Eg:

01.

6! ÷ 5!

6 × 5 × 4 × 3 × 2 × 1

5 × 4 × 3 × 2 × 1

This can be alternatively interpreted as

6! ÷ 5! =
6!
5!

=
6 × 5!
5!

= 6

Addition

Eg:

01.

5! + 6!	=	(5 × 4 × 3 × 2 × 1) + (6 × 5 × 4 × 3 × 2 × 1)
	=	120 + 720
	=	840

02.

5! + 6!	=	5! + 6 × 5!
	=	(1 + 6) × 5!
	=	7 × (5 × 4 × 3 × 2 × 1)
	=	7 × 120
	=	840

Subtraction

Eg:

01.

9! − 6!	=	(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) − (6 × 5 × 4 × 3 × 2 × 1)
	=	3,62,880 − 720
	=	3,62,160

02.

9! − 6!	=	9 × 8 × 7 × 6! − 6!
	=	(504 − 1) × 6!
	=	503 × (6 × 5 × 4 × 3 × 2 × 1)
	=	503 × 720
	=	3,62,160

LCM :: Explanation » Hide/Show

LCM is "Least Common Multiple".

Multiples

Multiples of a number are the successive products of the number and the natural numbers.

Eg:	01.	Multiples of 6 are 6 × 1, 6 × 2, 6 × 3,... ⇒ 6, 12, 18, ...

Common Multiples

Common multiples of two or more number are the multiples of the numbers which are common to all of them.

Eg:	01.	Multiples of 4 are — 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ..., 80, ..., 120, Multiples of 5 are — 5, 10, 15, 20, 25, 30, 35, 40, ..., 80, ..., 120, Multiples of 8 are — 8, 16, 24, 32, 40, ..., 80, ..., 120,

Eg:

01.

Multiples of 4 are — 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ..., 80, ..., 120,

Multiples of 5 are — 5, 10, 15, 20, 25, 30, 35, 40, ..., 80, ..., 120,

Multiples of 8 are — 8, 16, 24, 32, 40, ..., 80, ..., 120,

The common multiples of 4, 5, 8 are 40, 80, 120 ....

LCM is the least of the common multiples i.e. 40 ⇒ LCM of 4, 5, 8 is 40.

Notice the sequence of common multiples, the multiples of the LCM will also be the common multiples of the given numbers.

Factor

A factor of a given number is that number which divides it completely.

Eg:	01.	A factor of 24 is that number which divides 24 completely. ⇒ 1, 2, 3, 4, 6, 8, 12, 24 are factors of 24.

Eg:

01.

A factor of 24 is that number which divides 24 completely.

⇒ 1, 2, 3, 4, 6, 8, 12, 24 are factors of 24.

The following points are worth noting

The factors of a number include 1 and the number itself.
The factors of a number range between 1 and the number.
1 is the smallest factor of a given number (it is a factor of all the numbers).
The number itself is the highest factor of the given number.
The factor of a number divides the number completely.

Eg: 1 is a factor of 24. Therefore 1 divides 24 completely.
8 is a factor of 24. Therefore 8 divides 24 completely.

Eg:	1 is a factor of 24. Therefore 1 divides 24 completely. 8 is a factor of 24. Therefore 8 divides 24 completely.

If 'a' divides 'b' completely, then 'a' is a factor of 'b' and 'b' is a multiple of 'a'.

A number is a factor of its Multiple

Since a multiple is a product of the number and a natural number, the multiple of a number is always divisible by the number. Therefore we can say that a number is a factor of its multiple.

Test for LCM!!

LCM of a set of numbers is divisible by all the numbers in the set. If "a" is the LCM of "b", then "a" is divisible by "b".

To test whether a certain number is the LCM of two or more given numbers, we use this test of divisibility. If a number is the LCM of a set of numbers, then it should be divisible by all the numbers in the set.

However just because it is divisible we cannot say it is the LCM, since all the multiples of the LCM are also divisible by the given numbers.

Of all the numbers which are divisible by a set of numbers, the smallest one is the LCM. This can be interpreted as "LCM of a set of numbers is the smallest number divisible by all the numbers in the set."

The Highest of the given numbers can be the LCM

Since LCM should be divisible by all the given numbers whose LCM it is, it cannot be less than the highest of the given numbers. If at all one of the given numbers itself has a chance of being the LCM, it is the highest of the given numbers.

To test whether the highest of the given numbers is the LCM or not, divide it by the other numbers. If it is divisible by all of them completely then it is the LCM, otherwise not.

Since LCM ultimately should be a multiple of all the given numbers, if the highest number is not the LCM, then one of its multiples would be. This understanding can be used to calculate LCM orally for smaller numbers.

Eg:

LCM of 2, 4, 8:

Since 8 is the highest number, it can be the LCM.

Test whether 8 is divisible by the other numbers
8 is divisible by 2 completely.
8 is divisible by 4 completely.
⇒ 8 is divisible by all the given numbers.
Therefore 8 is the LCM of 2, 4, 8.

LCM of 3, 4, 8:

Since 8 is the highest number, it can be the LCM.

Test whether 8 is divisible by the other numbers
8 is not divisible by 3 completely.
⇒ 8 is not divisible by all the numbers.
Therefore 8 cannot be the LCM.

Consider the next multiple of 8 i.e 16
Test whether 16 is divisible by the other numbers
16 is not divisible by 3 completely.
Therefore 16 cannot be the LCM

Consider the next multiple of 8 i.e 24
Test whether 24 is divisible by the other numbers
24 is divisible by 3 completely.
24 is divisible by 4 completely.
⇒ 24 is divisible by all the given numbers.
Therefore 24 is the LCM of 3, 4, 8.

LCM of 4, 8, 14:

Since 14 is the highest number, it can be the LCM.

Test whether 14 is divisible by the other numbers
14 is not divisible by 4 completely.
⇒ 14 is not divisible by all the numbers.
[14 is not divisible by any of the other numbers]
Therefore 14 cannot be the LCM

Consider the next multiple of 14 i.e 28
Test whether 28 is divisible by the other numbers
28 is divisible by 4 completely.
28 is not divisible by 8 completely.
We need not check for divisibility by the highest number since this number is a multiple of the highest number.
Therefore 28 cannot be the LCM
[28 is divisible by 4 but not by 8]

We can consider the next multiple of 14 i.e 42 here.
Since 28 is divisible by 4 all its multiples would also be divisible by 4. Taking the multiples of 28 would eliminate the need to check for divisibility by 4].
To reduce your calculation, consider the next multiple of 28 i.e. 56.
[Since 28 is divisible by '14' all its multiples will be divisible by '14'.
You need not check for divisibility by '14' and "4" from here on, if you take multiples of 28.
Test whether 56 is divisible by the remaining other numbers
56 is divisible by 8 completely.
⇒ 56 is divisible by all the given numbers.
Therefore 56 is the LCM of 4, 8, 14

Eg:	LCM of 2, 4, 8: Since 8 is the highest number, it can be the LCM. Test whether 8 is divisible by the other numbers 8 is divisible by 2 completely. 8 is divisible by 4 completely. ⇒ 8 is divisible by all the given numbers. Therefore 8 is the LCM of 2, 4, 8. LCM of 3, 4, 8: Since 8 is the highest number, it can be the LCM. Test whether 8 is divisible by the other numbers 8 is not divisible by 3 completely. ⇒ 8 is not divisible by all the numbers. Therefore 8 cannot be the LCM. Consider the next multiple of 8 i.e 16 Test whether 16 is divisible by the other numbers 16 is not divisible by 3 completely. Therefore 16 cannot be the LCM Consider the next multiple of 8 i.e 24 Test whether 24 is divisible by the other numbers 24 is divisible by 3 completely. 24 is divisible by 4 completely. ⇒ 24 is divisible by all the given numbers. Therefore 24 is the LCM of 3, 4, 8. LCM of 4, 8, 14: Since 14 is the highest number, it can be the LCM. Test whether 14 is divisible by the other numbers 14 is not divisible by 4 completely. ⇒ 14 is not divisible by all the numbers. [14 is not divisible by any of the other numbers] Therefore 14 cannot be the LCM Consider the next multiple of 14 i.e 28 Test whether 28 is divisible by the other numbers 28 is divisible by 4 completely. 28 is not divisible by 8 completely. We need not check for divisibility by the highest number since this number is a multiple of the highest number. Therefore 28 cannot be the LCM [28 is divisible by 4 but not by 8] We can consider the next multiple of 14 i.e 42 here. Since 28 is divisible by 4 all its multiples would also be divisible by 4. Taking the multiples of 28 would eliminate the need to check for divisibility by 4]. To reduce your calculation, consider the next multiple of 28 i.e. 56. [Since 28 is divisible by '14' all its multiples will be divisible by '14'. You need not check for divisibility by '14' and "4" from here on, if you take multiples of 28. Test whether 56 is divisible by the remaining other numbers 56 is divisible by 8 completely. ⇒ 56 is divisible by all the given numbers. Therefore 56 is the LCM of 4, 8, 14

LCM of Factorials

If all the numbers are Factorials

If all the given numbers are factorials, then their LCM would be the highest of them

Eg:	01.	LCM of 6!, 8!, 24! is 24! [Since 24! is divisible by both 6! and 8!.] 24! = 24 × 23 × ... × 7 × 6! ⇒ 24! ÷ 6! = 24 × 23 × ... × 7 and 24! = 24 × 23 × ... × 9 × 8! ⇒ 24! ÷ 8! = 24 × 23 × ... × 9

Eg:

01.

LCM of 6!, 8!, 24! is 24! [Since 24! is divisible by both 6! and 8!.]

24! = 24 × 23 × ... × 7 × 6!
⇒ 24! ÷ 6! = 24 × 23 × ... × 7 and

24! = 24 × 23 × ... × 9 × 8!
⇒ 24! ÷ 8! = 24 × 23 × ... × 9

If all the numbers are not Factorials

If some of the given numbers are factorials, and one of the numbers representing the factorial ("n") is greater than all the other given numbers, then the LCM would be the highest of the factorials involved.

Eg:	01.	LCM of 5, 8, 6!, 10! is 10! [Since 10! is divisible by both 5 and 6!] 10! = 10 × 9 × ... × 6 × 5 × .. × 1 ⇒ 10! ÷ 5 = 10 × 9 × ... 6 × 4 × ... × 1 and 10! = 10 × 9 × ... × 7 × 6! ⇒ 10! ÷ 6! = 10 × 9 × ... × 7

Eg:

01.

LCM of 5, 8, 6!, 10! is 10! [Since 10! is divisible by both 5 and 6!]

10! = 10 × 9 × ... × 6 × 5 × .. × 1

⇒ 10! ÷ 5 = 10 × 9 × ... 6 × 4 × ... × 1 and

10! = 10 × 9 × ... × 7 × 6!

⇒ 10! ÷ 6! = 10 × 9 × ... × 7

If some of the given numbers are factorials, and the value of the factorial number is not the highest, then their LCM would have to be found out by evaluating the factorial value and adopting the normal procedure to find the LCM.

Eg: 01. LCM of 5, 8, 6!, is
6! = 6 × 7 × 5 × 4 × 3 × 2 × 1 ⇒ 6! = 720.
Thus LCM of 5, 8, 6!, ⇒ LCM of 5, 8, 720
Since 720 is divisible by 8 and 5, 720 is the LCM

Eg:	01.	LCM of 5, 8, 6!, is 6! = 6 × 7 × 5 × 4 × 3 × 2 × 1 ⇒ 6! = 720. Thus LCM of 5, 8, 6!, ⇒ LCM of 5, 8, 720 Since 720 is divisible by 8 and 5, 720 is the LCM

Author Credit : The Edifier ... Continued Page 3