We multiply the smallest powers of the common prime factors to obtain the HCF. Multiplying 3² = 9 and 5 = 5 gives 9 × 5 = 45.

find HCF of 135 and 225 by prime factorization

Answer: $\text{HCF}(135, 225) = 45$

Step-by-step solution

Step 1: Find the prime factorisation of each number

We express each number as a product of powers of its prime factors. Factoring $135$ , we divide by $3$ repeatedly to get $135 = 3 \times 3 \times 3 \times 5 = 3^3 \times 5$ . Factoring $225$ , we obtain $225 = 3 \times 3 \times 5 \times 5 = 3^2 \times 5^2$ .

Step 2: Identify the common prime factors and their smallest powers

By the Fundamental Theorem of Arithmetic, the $\text{HCF}$ is the product of the smallest power of each common prime factor involved in the numbers. The prime factors common to both $135$ and $225$ are $3$ and $5$ . The smallest power of $3$ is $3^2$ , and the smallest power of $5$ is $5^1$ .

Step 3: Calculate the HCF

We multiply the smallest powers of the common prime factors to obtain the $\text{HCF}$ . Multiplying $3^2 = 9$ and $5 = 5$ gives $9 \times 5 = 45$ .

Solve your own maths question free →