Skip to content
Exit presentation

A Square and A Cube

Grouping the primes in threes

A shop packs 2,744 barfi into a cube-shaped crate. How long is each edge, in barfi?

2,744 breaks into primes: 2 × 2 × 2 × 7 × 7 × 7.
Step 1 of 4
  1. 2,744 breaks into primes: 2 × 2 × 2 × 7 × 7 × 7.
  2. Groups of three: (2, 2, 2) and (7, 7, 7).
  3. Keep one from each: 2 and 7.
  4. 2 × 7 = 14. Check: 14 × 14 × 14 = 2,744.

Each edge is 14 barfi long.

Rule: Break the number into primes. Sort them into equal groups of three. Keep one prime from each group. Multiply what you kept.

The result, 14, is the cube root of 2,744. It is the edge of the cube. We write ∛2744 = 14.

Try 8,000. It has six 2s and three 5s. Make groups: (2, 2, 2), (2, 2, 2), (5, 5, 5). Keep 2, 2 and 5. 2 × 2 × 5 = 20. Check: 20 × 20 × 20 = 8,000.

Try 972. It is 2 × 2 × 3 × 3 × 3 × 3 × 3. Two 2s and two 3s are left over. So 972 is not a cube. Multiply 972 by 2 × 3 = 6 to get 5832. Now there are three 2s and six 3s. All fit in groups of three, so 5832 = 18³.

Tap Group in threes. See what is left over.

Loading lab…

Jump to lesson

This track

  1. 0 Start here
  2. 1 A number you can draw as a square
  3. 2 Each new layer is the next odd number
  4. 3 A test that can only say no
  5. 4 Going back to the side you started from
  6. 5 Pairing the primes finds the side
  7. 6 Checkpoint: squares, the staircase and roots
  8. 7 When no whole number fits
  9. 8 A cube number counts real cubes
  10. 9 What cubes end in, and what they add up to
  11. 10 Grouping the primes in threes
  12. 11 Which count is the job asking for
  13. 12 Checkpoint: in between, cubes and cube roots
  14. 13 The big picture
  15. 14 Track quiz

Other tracks