Rohan says 12 (= 2 × 2 × 3) must be a perfect square, since it breaks down cleanly into prime…

Why

Rohan has correctly broken 12 into its prime factors — 2 × 2 × 3 is exactly right, and knowing that every number breaks down cleanly into primes is genuinely the key idea behind this whole method. Where the rule needs one more condition is realising that breaking down cleanly isn't the same as being a perfect square. A perfect square's prime factorisation always has EVERY prime appearing an EVEN number of times — for example 36 = 2×2×3×3, where the 2 appears twice and the 3 appears twice, both even, letting the primes pair up perfectly (giving 6×6=36). Check 12's factorisation again: the 2 appears twice (even), but the 3 appears only ONCE (odd) — one prime is left over, unpaired, and that leftover 3 is exactly why 12 is not a perfect square (its square root, about 3.46, isn't a whole number). Test: 16 = 2×2×2×2, all four 2's, an even count — a genuine perfect square (4²). But 8 = 2×2×2, three 2's, an odd count — not a perfect square, despite factoring just as cleanly as 16 does. His teacher is right: every prime in the factorisation needs an EVEN count for a number to be a perfect square — factoring cleanly into primes at all is not enough by itself.