Three equal groups of primes
Triplets of prime factors make a cube.
Share the prime cards in three
A perfect cube is a number multiplied by itself three times. Look at
3375 = 3 × 3 × 3 × 5 × 5 × 5.
If 3375 is a number cubed, what would each of the three equal groups of primes need?
What this lesson covers
The idea
Each prime factor of a cube appears three times, so a number is a perfect cube if its prime factors can be split into three identical groups (or into triplets); the product of one group is its cube root.
Share the prime cards in three
A perfect cube is a number multiplied by itself three times. Look at 3375 = 3 × 3 × 3 × 5 × 5 × 5.
If 3375 is a number cubed, what would each of the three equal groups of primes need?
- One 3 and one 5
- Three 3s
- Two 5s
Three identical groups
Tap a card to move it between the pool and groups A, B and C. Make the three groups identical. If a number cannot be split, press the button!
Make triplets
A number is a perfect cube if its prime factors can be split into three identical groups (or into triplets). The product of one group is its cube root.
3375 = 3 × 3 × 3 × 5 × 5 × 5 = (3 × 5) × (3 × 5) × (3 × 5) = (3 × 5)^3 = 15^3, so ∛3375 = 15. As triplets: (3 × 3 × 3) × (5 × 5 × 5) = 3^3 × 5^3.
500 = 2 × 2 × 5 × 5 × 5. The two 2s do not make a triplet, so the factors cannot be split into three identical groups: 500 is not a perfect cube.
Each prime factor of a number appears three times as often in the prime factorisation of its cube.
Notes
A number is a perfect cube if its prime factors can be split into three identical groups (triplets). The product of one group is its cube root.
Check yourself
Which of these is a perfect cube?
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. Find its cube root.
Answer: 12
One group: 2 × 2 × 3 = 12, so 1728 = 12 × 12 × 12 and ∛1728 = 12.
20 = 2 × 2 × 5. How many times does the prime 2 appear in the prime factorisation of 20^3?
Answer: 6
20^3 = (2 × 2 × 5) × (2 × 2 × 5) × (2 × 2 × 5), so 2 appears 3 × 2 = 6 times.
500 = 2 × 2 × 5 × 5 × 5. Why is 500 not a perfect cube?
- 144 = 2 × 2 × 2 × 2 × 3 × 3. There are four 2s and only two 3s. Neither makes complete triplets: one 2 is left over and the 3s are short.
- 216 = 2 × 2 × 2 × 3 × 3 × 3 — correct. Yes! A triplet of 2s and a triplet of 3s: 216 = 6 × 6 × 6.
- 225 = 3 × 3 × 5 × 5. 3s and 5s come in pairs, not triplets. 225 = 15 × 15 is a square.
- 400 = 2 × 2 × 2 × 2 × 5 × 5. Four 2s and two 5s do not make complete triplets, so 400 is not a cube.
- Because 500 is even. Even numbers can be cubes: 8 = 2 × 2 × 2.
- Because 5 appears three times. Three 5s make a good triplet. The trouble is the 2s.
- Because the two 2s do not make a triplet — correct. Yes! A cube needs every prime in triplets, and 2 appears only twice.