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

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What is a Factorial?  
 

A factorial is a function whose domain is the set of whole numbers.

⇒ Factorials are defined for whole numbers only.
It is defined as (given by)   ∠n (Or) n! = n

k=1
k [n ≥ 0]
[∏ ⇒ Product ≡ Σ ⇒ Sum]

• Representing Factorials

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

k=1
k [n ≥ 0] , the range of values of "k" start with 1 and end with n.

Thus "k" would be equal to 1 both when "n" = 0 as well as when "n" = 1.

permutations,combinations,quantitative,techniques,methods,operations,research,linear,circular

Factorial Interpretation  
 
You will find factorials being used all throughout the topic permutations and combinations. It is a general idea which is also used in many other topics in mathematics. Knowing various ways in which the factorial may be interpreted would be useful in problem solving.

• 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).

• Factorial of a natural number

The factorial of a natural number "n" is the product of the all natural numbers less than or equal to "n".

⇒ Factorial "n" = 1 × 2 × 3 × ... × n.
⇒ n! or ∠n = 1 x 2 x 3 x 4 x ... x n.

Example

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 of 1 (One)

n! = 1 x 2 x 3 x 4 x ...x (n − 2) x (n − 1) x n. ⇒ 1! = 1

• Expanding factorials

n! or ∠n = 1 × 2 × 3 × 4 × ... × n.
(Or) = 1 × 2 × 3 × 4 × ... × (n − 2) x (n − 1) × n.
(Or) = n × (n − 1) × (n − 2) × (n − 3) × ... × 3 × 2 × 1.
(Or) = n × (n − 1)! [Since 1 x 2 x 3 x ... x (n - 1) = (n - 1)!]
(Or) = n × (n − 1) × (n − 2)! [Since 1 x 2 x 3 x ... x (n - 2) = (n - 2)!]
(Or) = 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.

Factorials » Arithmetic Operations  
 
permutations,combinations,quantitative,techniques,methods,operations,research,linear,circular

• 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!
=
6 × 5 × 4 × 3 × 2 × 1
5 × 4 × 3 × 2 × 1
= 6

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 of Factorials  
 

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,

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

Least Common Multiple

LCM of a set of numbers is the least of the common multiples of those numbers
In the above case LCM of 4, 5, 8 is 40 since 40 is the least of the common multiples.

The multiples of the LCM will also be the common multiples of the given numbers.
40, 80, 120 .... are the multiples of the LCM 40. They are also multiples of 4, 5 and 8.

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.

The following points are worth noting
  1. The factors of a number include 1 and the number itself.
  2. The factors of a number range between 1 and the number.
  3. 1 is the smallest factor of a given number (it is a factor of all the numbers).
  4. The number itself is the highest factor of the given number.
  5. 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.

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 "x" is the LCM of "a, b and c", then "x" is divisible by each of "a, b and c".

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.
[Of all the numbers which are divisible by the numbers in the set, only the least one is called the LCM.]

LCM of a set of numbers is the smallest number divisible by all the numbers in the set

The Highest of the given numbers may 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.

Example

  1. LCM of 2, 4, 8:
    1. Since 8 is the highest number, it may be the LCM.
    2. 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.

  2. LCM of 3, 4, 8:
    1. Since 8 is the highest number, it may be the LCM.
    2. 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 is not the LCM.

    3. 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 is not the LCM

    4. 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.

      We need not check for divisibility by the highest of the given numbers, since this number is a multiple of the highest number.

      Therefore 24 is the LCM of 3, 4, 8.

  3. LCM of 4, 8, 14:
    1. Since 14 is the highest number, it may be the LCM.
    2. 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 is not the LCM

    3. 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.

      Therefore 28 is not the LCM
      [28 is divisible by 4 but not by 8]

    4. We can consider the next multiple of 14 i.e 42 here.

      To reduce calculation we may follow this logic. 14 is not divisible by 4, but 28 which is a multiple of 14 is divisible by 4.

      All the multiples of 14 which are also multiples of 28 would only be divisible by 4.

      Since we are looking for multiples of 14 which are divisible by 4 apart from being divisible by 8 we may consider the multiples of 28 only from hereon.

      Taking the multiples of 28 would eliminate the need to check for divisibility by 4 as well as by 14.

      Consider the next multiple of 28 i.e. 56.

      Test whether 56 is divisible by the other remaining numbers

      56 is divisible by 8 completely.
      ⇒ 56 is divisible by all the given numbers.

      Therefore 56 is the LCM of 4, 8, 14

• Where all the numbers are Factorials

Where all the given numbers are factorials, their LCM would be the highest of them
Eg: 01. LCM of 6!, 8!, 24! is 24!
[Since all the numbers are factorials and 24! is the highest of them.]

Check

24!
6!
=
24 × 23 × ... × 7 × 6!
6!
= 24 × 23 × ... × 7 ⇒ 24! is divisible by 6!

24!
8!
=
24 × 23 × ... × 9 × 8!
8!
= 24 × 23 × ... × 9 ⇒ 24! is divisible by 8!

Thus 24! is the LCM of 6!, 8! and 24!

• Where 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! has got the highest numerical value (10) of all the given numbers]

    Check

    10!
    6!
    =
    10 × 9 × ... × 6!
    6!
    = 10 × 9 × 8 × 7 ⇒ 10! is divisible by 6!
    10!
    5
    =
    10 × 9 × ... × 5 ... × 2 × 1
    5
    = 10 × 9 × 8 × 7 × 6 × 4 × 3 × 2 × 1 ⇒ 10! is divisible by 5
    10!
    8
    =
    10 × 9 × 8 × ... × 2 × 1
    8
    = 10 × 9 × 7 × 6 × 5 × 4 × 3 × 2 × 1 ⇒ 10! is divisible by 8

    Thus 10! is the LCM of 5, 8, 6! and 10!

  • 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 ??
    ⇒ LCM of 5, 8, 720
    6! = 6 × 7 × 5 × 4 × 3 × 2 × 1
    = 720
    720
    8
    = 90 ⇒ 720 is divisible by 8
    720
    5
    = 144 ⇒ 720 is divisible by 5

    Thus 720 is the LCM of 5, 8 and 702 i.e. 5, 8 and 6!

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