Prime Factorization of 450

Break down $450$ into its prime factors: $450 = 45 \times 10 = (3^2 \times 5) \times (2 \times 5) = 2^1 \times 3^2 \times 5^2$.

$450 = 2^1 \times 3^2 \times 5^2$

Prime Factorization of 1500

Break down $1500$ into its prime factors: $1500 = 15 \times 100 = (3 \times 5) \times (2^2 \times 5^2) = 2^2 \times 3^1 \times 5^3$.

$1500 = 2^2 \times 3^1 \times 5^3$

Prime Factorization of 1650

Break down $1650$ into its prime factors: $1650 = 165 \times 10 = (3 \times 5 \times 11) \times (2 \times 5) = 2^1 \times 3^1 \times 5^2 \times 11^1$.

$1650 = 2^1 \times 3^1 \times 5^2 \times 11^1$

Determine HCF

Take the lowest power of each common prime factor among $450$, $1500$, and $1650$. The common prime factors are $2$, $3$, and $5$. Their lowest powers are $2^1$, $3^1$, and $5^2$.

$\text{HCF} = 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150$