Step 1: Prime Factorization of $310$

Express $310$ as a product of prime factors: $310 = 2 \times 155 = 2 \times 5 \times 31$.

$310 = 2 \times 5 \times 31$

Step 2: Prime Factorization of $208$

Express $208$ as a product of prime factors: $208 = 2 \times 104 = 2^2 \times 52 = 2^3 \times 26 = 2^4 \times 13$.

$208 = 2^4 \times 13$

Step 3: Prime Factorization of $180$

Express $180$ as a product of prime factors: $180 = 2 \times 90 = 2^2 \times 45 = 2^2 \times 3^2 \times 5$.

$180 = 2^2 \times 3^2 \times 5$

Step 4: Determine HCF

Find the lowest power of common prime factors among $310$, $208$, and $180$. The only common prime factor is $2$, and its lowest exponent across all three numbers is $1$. Thus, $\text{HCF} = 2^1 = 2$.

$\text{HCF}(310, 208, 180) = 2$