**Prime Factorization Of 84**

**What is Factors ?**

When two or more numbers are multiplied together to give a product, each is called a factor of the product. The product of two or more numbers is said to be a multiple of each of those numbers.

**For example 4 x 5 = 20. Here 4 and 5 are factors of 20 and 20 is a multiple of both 4 and 5. **

**OR**

**A number which divides the given number exactly ( without leaving any remainder ) is called a factor of given number.**

**Comparison Of Factors and Multiple**

Factors |
Multiples |

Every number is a factor of itself. | Every number is a multiple of itself. |

1 is a factor of every number. | 1 is a multiple of 1 only. |

The factors of a number are countable. | The multiple of a number are its uncountable. |

Every factor of a number other than the number itself is less than the number. | Every multiple of a number other than the number itself is greater than the number. |

**Factor :-**

**When two or more numbers are multiplied to give a product, each number is called a factor of the product. Here are the first eleven whole numbers and their factors:**

Whole Numbers |
Factors |

0 | 0,1,2,3,4,5,6,7,8,9………. |

1 | 1 |

2 | 1,2 |

3 | 1,3 |

4 | 1,2,4 |

5 | 1,5 |

6 | 1,2,3,6 |

7 | 1,7 |

8 | 1,2,4,8 |

9 | 1,3,9 |

10 | 1,2,5,10 |

**Thus, a number (called divisor) is a factor of another number (called dividend) if it divides the dividend exactly, ie there is no remainder left over. Prime Factorization Of 84**

**For example 3 is a factor of 36, but 3 is not a factor of 43.**

**3 IS Factor of 46**

**3 IS Not Factor of 46**

**PRIME FACTORIZATION**

**A prime number has two factors : itself and 1 while a composite number has more than two factors. Prime Factorization Of 84**

**The multiplication facts of 32 are as follows: **

**32 = 2x1x16**

**=2 x 1 x 2 x 2 x 2 x 2**

**= 2x2x2x2 x 2 x1**

**Usually we do not write the factor 1 as it is understood. Factors of 32 are thus 2, 2, 2, 2 and 2. Since all the 5 numbers are prime numbers these are called the prime factors of 32. We also note that a single prime number is repeated in the prime factorization of 32. Prime Factorization Of 84**

**Example : Give the prime factorization of 36.**

Start dividing 36 step by step by the prime numbers in order. First of all divide it by 2 as many times as possible. Thus,

**36** = **2 x 18 **

** = 2x2x9**

** = 2x2x3x3**

**Thus, the prime factors of 36 are 2,2,3,3.**

**Example** : **Find the prime factors of 64.**

** Use repeated division by prime numbers as it is the easiest way to get prime factors.**

**Prime Factorization Of 84**

The Factors of 84 are the number that produce the result as 84 when two numbers are multiplied together.

Factors of 84 are the whole numbers which could be either positive or negative but not a fraction or decimal number. **To find the factors of a number, 84 , we will use the factorization method. Prime Factorization Of 84**

**Factors of 84**

**1 , 2 , 3 , 4 , 6 , 7 , 12 , 14 , 21 , 28 , 42 , 84**

Pair Factors | Prime Factorization |

1×84 2×42 3×28 4×21 6×14 7×12 | 2×2×3×7OR 2 ^{2}×3×7 |

**In Factorization Method . first consider the number s. 1 and 84 and continue with finding the other pair of multiples of 84 , which gives the result as an original number. Prime Factorization Of 84**

**Pair Factors Of 84 **

1×84 = 84 ,then ( 1 , 84 ) is a pair factor 84

2×42 = 84 , then ( 2, 42 ) is a pair factor 84 ** { ∵42 = 2×3×7 }**

3×28 = 84 .then (3 , 28) is a pair of 84 **{ ∵28 = 2×14 }**

4×21= 84 .then ( 4 , 21 ) is a pair of 84 **{ ∵ 21 = 3×57 }**

6×14 = 84 .then ( 6 , 14) is a pair of 84 **{ ∵ 14 = 2×7 }**

7×12 = 84 .then ( 7 , 12) is a pair of 84 **{ ∵ 12 = 2×6 }**

42×2 = 84 , then ( 42 , 2 ) is a pair factor 84 ** { ∵42 = 2×3×7 }**

28×3 = 84 .then ( 23 ,3 ) is a pair of 84 **{ ∵28 = 2×14 }**

21×4= 84 .then ( 21 , 4 ) is a pair of 84 **{ ∵ 21 = 3×57 }**

14×6 = 84 .then ( 14 ,6 ) is a pair of 84 **{ ∵ 14 = 2×7 }**

12×7 = 84 .then ( 12,7 ) is a pair of 84 **{ ∵ 12 = 2×6 }**

**Here ( 2 , 42 ) is the same as ( 12 ,2 )**

**and (3 , 28 ) is the same as ( 28 , 3)**

**and (4, 21 ) is the same as ( 21 , 4 )**

**and (6 , 14 ) is the same as ( 14 , 6)**

**and (12 , 7 ) is the same as ( 7 , 12 )**

**The positive pair factor of 75 are ( 2 , 42 ) ,(3 , 28 ) , (4, 21 ) , (6 , 14 ) and (12 , 7 ) **

### Now we find the negative pair of factors , the proceed with following steps

**Pair Factors Of 84**

-1×-84 = 84 ,then (- 1 , -84 ) is a pair factor 84

-2×-42 = 84 , then ( -2, -42 ) is a pair factor 84 ** { ∵42 = 2×3×7 }**

-3×-28 = 84 .then (-3 , -28) is a pair of 84 **{ ∵28 = 2×14 }**

-4×-21= 84 .then ( -4 , -21 ) is a pair of 84 **{ ∵ 21 = 3×57 }**

-6×-14 = 84 .then ( -6 , -14) is a pair of 84 **{ ∵ 14 = 2×7 }**

-7×-12 = 84 .then ( -7 , -12) is a pair of 84 **{ ∵ 12 = 2×6 }**

-42×-2 = 84 , then ( -42 , -2 ) is a pair factor 84 ** { ∵42 = 2×3×7 }**

-28×-3 = 84 .then ( -23 ,-3 ) is a pair of 84 **{ ∵28 = 2×14 }**

-21×-4= 84 .then ( -21 , -4 ) is a pair of 84 **{ ∵ 21 = 3×57 }**

-14×-6 = 84 .then (- 14 ,-6 ) is a pair of 84 **{ ∵ 14 = 2×7 }**

-12×-7 = 84 .then ( -12, -7 ) is a pair of 84 **{ ∵ 12 = 2×6 }**

**Here (- 2 , -42 ) is the same as (- 12 ,-2 )**

**and (-3 , -28 ) is the same as (- 28 ,- 3)**

**and (-4, -21 ) is the same as ( -21 ,- 4 )**

**and (-6 , -14 ) is the same as (- 14 ,- 6)**

**and (-12 , -7 ) is the same as (- 7 ,- 12 )**

**Now we calculate the Factors of 84?**

**Following steps to calculate the factors of a number.**

First, write the number 84. Prime Factorization Of 84

Find the two numbers, which gives the result as 84 under the multiplication, 2 and 42, such as 2 x 42 = 84.

We know that 2 is a prime number which has only two factors, i.e., 1 and the number itself( 1 and 2).

2 =2 x 1

But look at the number 42, which is a natural number but not a prime number. So it can be further factorized

42 = 2 x 3 x7x1

Therefore, the factorization of 84 is written as, 84 = 2×2×3×7x 1

Finally, write down all the unique numbers (i.e.) 2×2×3×7x 1

Factors of 84 |

1 , 2 , 3 , 4 , 6 , 7 , 12 , 14 , 21 , 28 , 42 and 84 |

**Prime Factors Of 84 BY Using Division Method**

**First Step :- ****Divide the number 84 with the smallest prime factor , say 2. **

**Second Step :- ****84 ÷ 2 = 42**

**Know again divide 42 by 2**

**42 ÷ 2 = 21**

**Third Step :- If we divide 21 by 2 we will get a fractional number ,which cannot be a factor.**

**Fourth step :- Now proceed with next prime numbers , i;e 21 ÷ 3 = 7 . **

**Fifth Step :- If we divide 7 by 3 we will get a fractional number , which cannot be a factor. Prime Factorization Of 84**

**Sixth Step :-** **Now proceed with next prime numbers, i;e 7 ÷ 7 = 1**

** Now we received prime factors of 84 are 2×2×3×7 OR 2 ^{2}×3×7^{ }**

**Thus , the Prime Factors of 84 are **** = 2×2×3×7**

** = 2**^{2}×3×7

^{2}×3×7