What is Prime Factorization, And How Does It Work?

The factor of a given number that is a prime number is called a prime factor. Factors are numbers that are multiplied together to produce a new number. To put it another way, prime factoring is the process of determining which prime numbers multiply to produce the original number.

Breaking a number down into a series of prime numbers that multiply to produce the original number is known as prime factorisation or integer factorisation. Prime decomposition is another name for this.

Example: The prime factors of 21 are 3 and 7 (because 3 × 7 = 21, and 3 and 7 are prime numbers).

Here we will find the factor pairs of 120 through prime factorisation.

Definition

All integers (positive and negative whole numbers) that may be evenly divided into 120 are called Factors of 120. A Factor of 120 divided by another Factor of 120 equals another Factor of 120.

Prime Factorization Methods

The following are the most widely used prime factorisation methods:

Division Method

The steps for calculating a number’s prime factors are identical to those for determining the factors of a large number.

To find the prime factors of an integer using the division method, follow the steps below:

Step 1: Take the smallest prime number and divide it by the specified number. The smallest prime number should split the integer exactly in this scenario.

Step 2: Subtract the quotient from the smallest prime number once more.

Step 3: Continue until the quotient is equal to one.

Step 4: Finally, add all of the prime factors together.

Example

Using 120 as an example, the following is a detailed step-by-step technique of prime factorisation.

Step 1: Divide 120 by the least prime number, which is 2.

As a result, 120 ÷ 2 = 60.

Step 2: Once more, divide 60 by the least prime number (which is again 2).

60 ÷ 2 = 30 now.

Step 3: Again divide 30 by the least prime number

30 ÷ 2 = 15 now

Step 4: Divide by the smallest prime number, which is 3.

As a result, 15 ÷ 3 = 5.

Step 5: Because 5 is a prime number, divide it by one.

120’s prime factors will now be 23x 3 x 5.

Factor Tree Method

Follow the steps below to use the factor tree method to get the prime factorisation of a given number:

Step 1: Think of the given number as the tree’s root.

Step 2: Draw a tree with the pair of components as branches.

Step 3: Factorise the composite factors once more and write the factor pairs down as branches.

Step 4: Repeat step until all of the composite factors’ prime factors have been found.

The factors of a number are located in a factor tree, and then those numbers are further factorised until the closure is reached.

Let’s say we need to use a factor tree to discover the factors of 120

We may factorise the number 120 into two numbers, namely 10 and 12.

Again, 

10 and 12 is factorised to get the prime factors of 10 and 12, such that;

10 = 2 x 5

and 

12 = 2 x 2 x 3

If we write the prime factors of 120 altogether, then;

120 = 10 x 12

       = 2 x 2 x 2 x 3 x 5

How do I find the 120 Factors?

Because the Factors of 120 are all the numbers that divide evenly into 120, we have to divide 120 by all the numbers up to 120 to determine which ones produce an even quotient. We discovered that these calculations resulted in an even quotient when we did so:

120 ÷ 1= 120
120 ÷ 2= 60
120 ÷ 3= 40
120 ÷ 4= 30
120 ÷ 5= 24
120 ÷ 6= 20
120 ÷ 8= 15
120 ÷ 10= 12
120 ÷ 12= 10
120 ÷ 15= 8
120 ÷ 20= 6
120 ÷ 24= 5
120 ÷ 30= 4
120 ÷ 40= 3
120 ÷ 60= 2
120 ÷ 120= 1

As a result, the Positive Factors of 120 are all the numbers we divided (divisors) to get an even number. The following is a list of all 120 Positive Factors in numerical order:

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

Negative numbers are included in factors of 120. As a result, all of 120’s Positive Factors can be turned into negative numbers. The following are the 120 Negative Factors:

-1, -2, -3, -4, -5, -6, -8, -10, -12, -15, -20, -24, -30, -40, -60, and -120.

Prime Factors of 120 are 2, 3, and 5

Prime Factorization of 120: 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5

Sum of Factors of 120: 360

What is the total number of Factors of 120?

We discovered that 120 has 16 positive and 16 Negative Factors when we counted the abovementioned factors. As a result, there are 32 Factors in all.

Factor Pairs of 120

Factor pairs of the number 120 are made up of two factors that add up to 120 when multiplied together. Here are all of the 120 Positive Factor Pairs.

The product that Produces 120ResultFactor Pairs
1 × 120= 1201;120
2 × 60= 1202;60
3 × 40= 1203;40
4 × 30= 1204;30
5 × 24= 1205;24
6 × 20= 1206;20
8 × 15= 1208;15
10 × 12= 12010;12
12 × 10= 12010;12
15 × 8= 1208;15
20 × 6= 1206;20
24 × 5= 1205;24
30 × 4= 1204;30
40 × 3= 1203;40
60 × 2= 1202;60
120 × 1= 1201;120

Negative numbers are included in factors of 120. Because minus times minus equals plus, you may convert the Positive Element Pair list above to all the negative factor pairs of the number 120 by simply adding a minus in front of each factor.

-1 × -120= 120
-2 × -60= 120
-3 × -40= 120
-4 × -30= 120
-5 × -24= 120
-6 × -20= 120
-8 × -15= 120
-10 × -12= 120
-12 × -10= 120
-15 × -8= 120
-20 × -6= 120
-24 × -5= 120
-30 × -4= 120
-40 × -3= 120
-60 × -2= 120
-120 × -1= 120

(-1;-120), (-2;-60), (-3;-40), (-4;-30), (-5;-24), (-6;-20), (-8;-15) and (-10;-12) are the negative pair factors of 120.

Important Points to know

Keep the following in mind while determining a number’s factors:

  • A number’s factors are always 1 and the number itself.
  • To find the number’s additional components, we must first determine its prime factorisation. The prime factors of the number are therefore multiplicands of the prime factorisation.
  • The composite factors of a number are obtained by multiplying some or all multiplicands in various combinations.
  • Any number cannot have fractions or decimals that are not integers as factors.
  • The additive inverse of a number that is a factor of the provided number is also a factor of the given number.

Because 8 is a factor of 120, -8 is a factor of 120 as well.

Conclusion

Prime factorisation is essential in mathematics for determining the HCF and LCM and the type of decimal expansion of any rational number. At Vedantu, we begin at the very beginning so that students can learn from the ground up. Pupils receive summary notes on subjects, doubt clearing sessions, parental consultation, and a variety of additional elements that help students develop holistically.

FAQ

What are the most important Factorisations?

Prime factorisation is a list of prime numbers that produce a specific product when multiplied.

What is the prime factorisation of 120? Find using a long division method.

We can utilise the long division approach, as shown below, to obtain the prime factors of 120.

What is the sum of all the factors in the number 120?

Sum of all factors of 120 = (23 1 – 1)/(2 – 1) × (31 1 – 1)/(3 – 1) × (51 1 – 1)/(5 – 1) = 360

What is the biggest common factor between the numbers 120 and another?

The greatest common factor can be calculated by comparing the prime factorisation (in some books) of two numbers and obtaining the highest common prime factor. The GCF is 1 if there is no common factor. This is also known as the highest common factor and is the common prime factor of one of the two numbers. It is the prime factor that the two numbers share as the greatest factor (largest number). Any pair of integers have one least common factor (the smallest number in common).

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