Distinct Prime Factors of a Number
The distinct primes of a number are the different prime numbers that occur in the factorization of that number. For example, the prime factorization of 20 is 2 × 2 × 5. Here the distinct prime factors of 20 are 2 and 5.
Let us recap our basics about the factors of a number.
What are the factors of any number?
Any number can be a factor of a number if it divides the number without leaving any remainder behind i.e., the remainder is zero. For example, 2 is a factor of 8, 7 is a factor of 42, etc.
What are the prime factors of any number?
Prime numbers are those numbers that do not have factors other than 1 and itself. Factors of a number that are prime themselves are prime factors.
For example, the factors of 20: 1, 2, 4, 5, 10 and 20. Here, 2 and 5 are prime numbers. Hence they are prime factors of 20.
How to determine Prime Factors of any number?
We divide the number by prime numbers in this method. In this method, we will continue division with the quotient if it’s a composite number till it doesn’t leave a remainder. The prime factors can be obtained by two methods which are:
(1) Division Method
(2) The Factor Tree Method
1. Prime Factorization by Division Method
We divide the number by the smallest prime number which doesn’t leave a remainder. The quotient obtained is divided repeatedly by the smallest prime number until the last quotient obtained is 1.
For example, let us factorize the number 12.
Divide 12 by the prime number 2
12 ÷ 2 = 6, and continue similarly
6 ÷ 2 = 3
3 ÷ 3 = 1
Hence the prime factorization of 12 is 2×2×3.
2. Prime Factorization by the Factor Tree Method
We need to find the prime factorization hence, the root of the factor tree is the number itself. We can take the pair of factors as the branch of the number. The factor tree ends at a prime factor.
For example, let us factorize the number 30.
The factor tree ends at 5 since it is a prime number.
∴ prime factorization of 30 is 2 × 3 × 5.
Here, the distinct prime factors of 30 are 2, 3, and 5.
It becomes easy to determine the distinct prime factors of a number after visualising its prime factorisation.
Examples
Example 1: What are the distinct prime factors of the number 24?
Solution: Let’s factorize 24 by division method:
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
Hence the prime factorization of 24 is 23 × 3.
∴ The distinct primes of 24 are 2 and 3.
Example 2: Factorize 32 by factor tree method and division method.
Solution: Factorization of 32 by factor tree method is:
Hence the prime factorization of 32 is 2 \times 2 \times 2 \times 2 \times 2=2^{5}
Now factorizing again by division method:
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
Hence the prime factorization of 32 is 2 \times 2 \times 2 \times 2 \times 2=2^{5}
The distinct prime factor of 32 is 2.
