Find the prime factorisation of the following numbers: (1) 72 (ii) 172 (iii) 450 (0) 8712 (vi) 13500 (iv) 980
The Correct Answer and Explanation is:
Here is the prime factorisation of the given numbers:
- 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
- 172 = 2 × 2 × 43 = 2² × 43
- 450 = 2 × 3 × 3 × 5 × 5 = 2 × 3² × 5²
- 8712 = 2 × 2 × 2 × 3 × 11 × 33 = 2³ × 3 × 11 × 33
- 980 = 2 × 2 × 5 × 7 × 7 = 2² × 5 × 7²
- 13500 = 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 = 2² × 3³ × 5³
Explanation
Prime factorisation is the process of breaking down a composite number into a product of prime numbers. Prime numbers are numbers greater than 1 that have only two positive divisors — 1 and the number itself. Examples of prime numbers include 2, 3, 5, 7, 11, and 13.
To factorise a number into primes, we begin by dividing the number by the smallest prime, which is 2, and continue dividing until it is no longer divisible by that prime. We then move to the next smallest prime and repeat the process. This continues until we are left with 1.
Let’s take 72 as an example. Dividing by 2, we get:
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 is not divisible by 2, so we move to 3:
9 ÷ 3 = 3
3 ÷ 3 = 1
This gives the prime factorisation 2³ × 3².
For 450, we divide:
450 ÷ 2 = 225
225 ÷ 3 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
So the prime factorisation is 2 × 3² × 5².
This process helps identify the building blocks of a number and is useful in simplifying fractions, finding the greatest common divisor, least common multiple, and in various applications in algebra and number theory.
Every whole number greater than 1 either is a prime or can be written uniquely as a product of prime numbers, which is known as the Fundamental Theorem of Arithmetic.
