Prime Factorization of the numbers

Express each number as a product of prime factors: $240 = 2^4 \times 3^1 \times 5^1$, $360 = 2^3 \times 3^2 \times 5^1$, and $480 = 2^5 \times 3^1 \times 5^1$.

$240 = 2^4 \times 3^1 \times 5^1, 360 = 2^3 \times 3^2 \times 5^1, 480 = 2^5 \times 3^1 \times 5^1$

Identify common prime factors

To find the HCF, take the lowest exponent of every common prime factor present in all three numbers.

$\text{HCF} = 2^{\min(4,3,5)} \times 3^{\min(1,2,1)} \times 5^{\min(1,1,1)}$

Calculate the result

Multiply the lowest powers found: $2^3 = 8$, $3^1 = 3$, $5^1 = 5$. Thus, $8 \times 3 \times 5 = 120$.

$\text{HCF} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$