Break a number into primes
84 laddus are packed into equal boxes. Nothing is left over. Which box sizes work?
- Start at 84.
- Divide by 2: 42.
- Divide by 2 again: 21.
- Divide by 3: 7.
- Divide by 7: 1. Primes: 2 × 2 × 3 × 7.
A box size works when it divides 84. The primes of 84 show every such size.
The rule
A factor of a number divides it with nothing left over. A prime has exactly two factors. A composite has more than two.
Divide by the smallest prime that fits. The number gets smaller each time. So the dividing must stop.
Keep dividing by primes until you reach 1.
Every composite number is a product of primes. That product is its prime factorisation.
Box sizes
84 = 2 × 2 × 3 × 7. A box size is a product of some of these primes. Each prime can be used no more times than it appears.
So 4 works, because 4 = 2 × 2. A box of 5 never works. 5 is not a prime factor of 84.
Is it composite?
Is 5 × 7 × 11 + 7 composite? Both parts have the factor 7. So it is 7 × (5 × 11 + 1), which is 7 × 56. It has factors 7 and 56. So it is composite.
Now try it. Divide by primes until you reach 1. A prime that does not divide is refused. The box sizes of 84 light up as you go.
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