Checking converses

To test a converse:

• write it out in full;
• hunt for a counterexample;
• if none turns up, try to prove it.

Divisibility traps. Being divisible by 4 and by 6 only guarantees divisibility by their LCM, 12, not by 24: 12 itself is a counterexample. But 5 and 12 share no factor, so divisible by both does mean divisible by 60.

Squares lose the sign; cubes keep it. x² = y² does not mean x = y (2² = (–2)²). But x³ = y³ does force x = y for real numbers, because 2³ = 8 and (–2)³ = –8.

Counting factors:

• p² (p prime) has exactly 3 factors, and having exactly 3 factors forces a prime squared: both directions hold;
• p × q has 4 factors, but 4 factors can also come from p³: 8 has 1, 2, 4, 8.

Divisible by 8 ⇒ divisible by 2 and 4, but not the other way: 12 is divisible by 2 and by 4, but not by 8.

A subtle one (starred, enrichment). If AB = AC, the bisector lengths IE and IF are equal. The converse fails: in any triangle with ∠A = 60°, IE = IF even when AB ≠ AC.

'Enough' versus 'needed'. 'Divisible by 8' is enough to make a number divisible by 4; it is not needed (12 is divisible by 4 without it). A converse asks whether the 'enough' condition is also 'needed'.