Step 1: Find the prime factorisation of 408.
$408 = 2 \times 204 = 2 \times 2 \times 102 = 2 \times 2 \times 2 \times 51 = 2^3 \times 3 \times 17$
Step 2: Find the prime factorisation of 312.
$312 = 2 \times 156 = 2 \times 2 \times 78 = 2 \times 2 \times 2 \times 39 = 2^3 \times 3 \times 13$
Step 3: Find the H.C.F. (Highest Common Factor).
H.C.F. = product of the lowest powers of all common prime factors. Common primes: 2 (power 3) and 3 (power 1). \[ \text{H.C.F.} = 2^3 \times 3 = 8 \times 3 = 24 \]
Step 4: Find the L.C.M. (Least Common Multiple).
L.C.M. = product of the highest powers of all prime factors. \[ \text{L.C.M.} = 2^3 \times 3 \times 13 \times 17 = 8 \times 3 \times 13 \times 17 \]
Step 5: Calculate the L.C.M.
$8 \times 3 = 24$, $24 \times 13 = 312$, $312 \times 17 = 5304$. So L.C.M. = 5304. Verify: $\text{H.C.F.} \times \text{L.C.M.} = 24 \times 5304 = 127296 = 408 \times 312$. Check!
Step 6: Conclusion.
$\text{H.C.F.} = 24$ and $\text{L.C.M.} = 5304$.
\[ \boxed{\text{H.C.F.} = 24, \quad \text{L.C.M.} = 5304} \]