Squares have even powers
36 dots can fill a square rangoli grid. 40 dots cannot.
- 12 breaks into 2 × 2 × 3.
- Square it: 12 × 12 = 144.
- Every power doubles: 2 × 2 × 2 × 2 × 3 × 3.
- Written short: 144 = 2⁴ × 3².
36 = 6 × 6, so 6 rows of 6 fill the grid. 40 is not a whole number times itself.
The rule
The power of a prime is how many times it appears. A perfect square is a whole number times itself.
Take a whole number n. Squaring doubles the power of every prime. That is because n × n uses each prime of n twice. So:
A perfect square has an even power of every prime.
The reverse is true too. A number with every power even is a perfect square. If every power is even, pair the primes up. Each pair gives one prime of the square root.
| Number | Prime factors | Powers | Perfect square? |
|---|---|---|---|
| 36 | 2 × 2 × 3 × 3 | 2², 3² | yes, 6 × 6 |
| 40 | 2 × 2 × 2 × 5 | 2³, 5¹ | no, its powers are odd |
| 49 | 7 × 7 | 7² | yes, 7 × 7 |
| 72 | 2 × 2 × 2 × 3 × 3 | 2³, 3² | no, 2 has an odd power |
Make a square
Multiply 72 by the smallest number that makes a perfect square. What is it? 72 = 2³ × 3². Only the power of 2 is odd. Multiply by 2. You get 144 = 12 × 12.
Now try it. Slide the number. Look at each power.
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