Prime factorization means breaking a number into smaller prime numbers. As we know, a prime number is a whole number that is only divisible by 1 and the number itself.
These numbers are called factors, which, when multiplied, give back the original number.
Prime Factors
Here, since 3 is a prime number, it is a prime factor, whereas 4 is a composite number, which is further factored to get 2 prime factors (2 and 2).
Thus, 3, 2, and 2 are the prime factors obtained after the prime factorization of 12.
All such prime numbers obtained after prime factorization are called prime factors. In prime factorization, we continue factoring until we get all the prime factors of a number.
Here are a few more examples of prime factors obtained after prime factorization.
Number
In Exponent Form
In Expanded Form
72
23 × 32
2 × 2 × 2 × 3 × 3
36
22 × 32
2 × 2 × 3 × 3
30
2 × 3 × 5
2 × 3 × 5
48
24 × 3
2 × 2 × 2 × 2 × 3
35
5 × 7
5 × 7
63
32 × 7
3 × 3 × 7
42
2 × 3 × 7
2 × 3 × 7
Applications
It has applications in:
Simplifying fractions
Finding the greatest common divisor (GCD)
Determining the least common multiple (LCM) and
Cryptography
Prime factorization can be done in two ways: the Division Method and the Factor Tree Method.
By Factor Tree Method
In this method, we factorize a number into any two factors and continue factoring each non-prime number into smaller factors until all the branches end with prime numbers.
Let us find the factors of 56 by the factor tree prime factorization method.
Step 1: Placing the Number on the Top of the Tree
Step 2: Splitting the Number into Two Factors
56 can be factored as 7 × 8
Now, the factors are placed just below the branches of the tree, as shown:
Here, 56 = 7 × 8
Step 3: Continue to Split Each Composite Factor
Now, we split the composite factor further
8 = 2 × 4
Step 4: Repeating Until All Factors Are Prime
Repeating the process until all branches end in prime numbers,
4 = 2 × 2
All the divisors used are the prime factors of the original number.
Thus, the prime factorization of 56 is 56 = 23 × 7. The factor tree will look like this:
Often for representation, all prime numbers in a factor are represented by a circle, and the composite numbers that are further factored are shown using square boxes as shown below:
Note: No other combination of prime numbers can produce 56, and every number has its unique set of prime factors
For example, the two different factor trees of 56 are given below.
By Division Method
This method involves repeatedly dividing the number by the smallest possible prime numbers until the quotient becomes 1
Let us find the factors of 78 by the division method.
Step 1: Dividing the Number by the Smallest Prime Number
We begin by dividing with the smallest prime number (usually 2). If not, we then proceed with the next smallest prime number.
Here, the quotient is 39, which is not divisible by 2
Step 2: Continuing to Divide By Prime Numbers Until the Quotient Becomes 1
We then continue to divide by the same prime number until it is no longer divisible by it. We then proceed to the next smallest prime number.
Now, Dividing 39 by the prime number 3,
39 ÷ 3 = 13
Here, the quotient is 13, a prime. Now, 13 ÷ 13 = 1
Thus, the prime factorization of 78 is 78 = 2 × 3 × 13
Solved Examples
Find the prime factorization of 198.
Solution:
Thus, the prime factorization of 198 is 198 = 2 × 32 × 11
Find the prime factorization of the following numbers using the factor tree method. a) 75 b) 84 c) 24 d) 18
Solution:
Thus, the prime factorizations are: a) 75 = 3 × 52 b) 84 = 22 × 3 × 7 c) 24 = 23 × 3 d) 18 = 2 × 32
Find the prime factorization of the following numbers using the division method. a) 100 b) 54 c) 96 d) 27
Solution:
Thus, the prime factorizations are: a) 100 = 22 × 52 b) 54 = 2 × 33 c) 96 = 25 × 3 d) 27 = 33
Write 224 as a product of its prime factors.
Solution:
Thus, 224 = 25 × 7
What Number Under 100 Has the Most Prime Factors?
Solution:
96 = 25 × 3 Thus, 96 has the most prime factors.
How many different prime factors does 48 have?
Solution:
Here, 48 has 2 different prime factors, i.e., 2 and 3
