When both directions hold

Factors come in partner pairs whose product is the number. For 12: (1, 12), (2, 6), (3, 4). Six factors, an even count.

One direction: an odd number of factors ⇒ a perfect square. If the count is odd, some pair must repeat a number, (f, f). Then the number is f × f: a perfect square.

The other direction: a perfect square ⇒ an odd number of factors. This needs one more fact: only one pair can repeat, because f must be √n. So exactly one factor is counted once and the rest come in twos. For 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) gives 9 factors.

Both directions hold, so 'n is a perfect square' and 'n has an odd number of factors' say the same thing. Each implies the other.

The converse of the Baudhāyana–Pythagoras theorem is true too: if a² + b² = c², the triangle has a right angle. Build a right triangle XYZ with legs a and b; its hypotenuse is √(a² + b²) = c. The two triangles have the same three sides, so they are congruent (SSS), and the original has a right angle too.

The divisibility rule for 3 works both ways: a number is divisible by 3 if and only if its digit sum is a multiple of 3.