Step 1: Choose a different route to the HCF.
Instead of factorizing both numbers into primes, let us use Euclid's division algorithm, which finds the HCF by repeated division. The rule is: for two positive integers $a$ and $b$ with $a \gt b$, divide $a$ by $b$ to get $a = bq + r$. The HCF of $a$ and $b$ equals the HCF of $b$ and $r$. We repeat until the remainder becomes 0, and the last non-zero divisor is the HCF.
Step 2: First division.
Take $a = 432$ and $b = 96$.
\[ 432 = 96 \times 4 + 48 \]
Here the quotient is 4 and the remainder is 48.
Step 3: Second division.
Now divide 96 by the remainder 48.
\[ 96 = 48 \times 2 + 0 \]
The remainder has become 0.
Step 4: Read off the HCF.
Since the remainder is 0 at this stage, the divisor used just before, namely 48, is the HCF of 96 and 432. This matches the value we would also get from prime factorization, since $2^4 \times 3 = 48$.
Final Answer:
The HCF of 96 and 432 is 48, which corresponds to option (A).
\[ \boxed{48} \]