Pairing the primes finds the side
Trying sides one by one for 1764 is slow. Primes give a method.
- 36 breaks into primes: 2 × 3 × 2 × 3.
- Sort into pairs: (2, 2) and (3, 3).
- Keep one from each: 2 and 3.
- 2 × 3 = 6. Check: √36 = 6.
Rule: Break the number into primes. Sort them into equal pairs. Keep one prime from each pair. Multiply what you kept.
For 1764, the primes are 2 × 2 × 3 × 3 × 7 × 7. Keep 2, 3 and 7. 2 × 3 × 7 = 42.
A square is a side times itself. Each prime of the side appears twice. So the primes of a square come in pairs. If one prime has no partner, the number is not a square.
Try 400. It has four 2s and two 5s. The pairs are (2, 2), (2, 2) and (5, 5). Keep 2, 2 and 5. 2 × 2 × 5 = 20. Check: 20 × 20 = 400.
Try 132. It is 2 × 2 × 3 × 11. The 3 and the 11 have no partner. So 132 is not a square. Multiply 132 by 3 × 11 = 33 to get 4356. Now the primes are 2 × 2 × 3 × 3 × 11 × 11. Each has a partner, so 4356 = 66 × 66.
Tap Pair them up. See what is left alone.
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