A Square and A Cube · Lesson 2 of 4
Cubic Numbers
“Explore cubic numbers, taxicab numbers, odd-number patterns, cube roots and successive differences.”
• Explain a cubic number using unit cubes and repeated multiplication. • Recognise common perfect cubes and patterns in their last digits. • Describe the Hardy–Ramanujan number 1729 and the idea of taxicab numbers. • Connect perfect cubes with groups of consecutive odd numbers. • Find cube roots using known cubes and prime factorisation in triplets.
A cube is a three-dimensional solid with equal edge lengths. If its edge is 3 units, one layer contains 3 × 3 = 9 unit cubes, and there are 3 such layers. So the full cube contains 3 × 3 × 3 = 27 unit cubes.
A number obtained by multiplying a whole number by itself three times is called a cube. Cubes of natural numbers are called perfect cubes.
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
The last digit of a cube depends only on the last digit of the original number. Unlike perfect squares, perfect cubes can end in any digit from 0 to 9. For example, numbers ending in 2 have cubes ending in 8, while numbers ending in 8 have cubes ending in 2.
| Units digit of a number | Units digit of its cube |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 7 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 3 |
| 8 | 2 |
| 9 | 9 |
Trailing zeros follow a simple cube pattern. If a whole number ends in one zero, its cube ends in three zeros; if it ends in two zeros, its cube ends in six zeros. Therefore, a perfect cube cannot end in exactly two zeros.
Problem
Find (−6)³.
- 1.(−6)³ = (−6) × (−6) × (−6).
- 2.The first two negative factors give +36.
- 3.36 × (−6) = −216.
- 4.Therefore, (−6)³ = −216.
Problem
Find (4/6)³.
- 1.Cube the numerator and denominator.
- 2.(4/6)³ = 4³/6³ = 64/216.
- 3.The fraction can also be simplified to 8/27.
Taxicab Numbers
The number 1729 became famous through an exchange between Srinivasa Ramanujan and G. H. Hardy. Ramanujan pointed out that 1729 is the smallest number that can be written as the sum of two positive cubes in two different ways.
In this chapter, taxicab numbers are numbers that can be expressed as the sum of two positive cubes in two different ways.
Problem
Verify both cube-sum representations of 1729.
- 1.1³ + 12³ = 1 + 1728 = 1729.
- 2.9³ + 10³ = 729 + 1000 = 1729.
- 3.Both different pairs give the same total.
Perfect Cubes and Consecutive Odd Numbers
Perfect cubes can also be built from groups of consecutive odd numbers. The groups grow in size: one odd number gives 1³, two consecutive odd numbers give 2³, three give 3³, and so on.
| Consecutive odd numbers | Sum | Cube |
|---|---|---|
| 1 | 1 | 1³ |
| 3 + 5 | 8 | 2³ |
| 7 + 9 + 11 | 27 | 3³ |
| 13 + 15 + 17 + 19 | 64 | 4³ |
| 21 + 23 + 25 + 27 + 29 | 125 | 5³ |
Problem
Find 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 without adding one term at a time.
- 1.There are 10 consecutive odd numbers in this group.
- 2.In the pattern, the group containing 10 odd numbers sums to 10³.
- 3.Therefore, the sum is 1000.
Cube Roots
Cube root reverses cubing. Since 2³ = 8, the cube root of 8 is 2. Since 10³ = 1000, the cube root of 1000 is 10.
If y = x³, then x is the cube root of y. We write this as ∛y = x.
Prime factorisation gives a clear cube test. In a perfect cube, equal prime factors can be grouped completely into triplets. Take one factor from each triplet to get the cube root.
Problem
Find ∛3375.
- 1.3375 = 3 × 3 × 3 × 5 × 5 × 5.
- 2.Group the factors into triplets: (3 × 3 × 3)(5 × 5 × 5).
- 3.Take one factor from each triplet: 3 × 5 = 15.
- 4.Therefore, ∛3375 = 15.
Problem
Use prime factorisation to decide whether 500 is a perfect cube.
- 1.500 = 2 × 2 × 5 × 5 × 5.
- 2.The three 5s form one triplet, but the two 2s cannot form a complete triplet.
- 3.Therefore, 500 is not a perfect cube.
Successive Differences
For perfect squares, taking differences twice gives a constant pattern. For perfect cubes, we continue one level further. Using 1, 8, 27, 64 and 125, the first differences are 7, 19, 37 and 61; the second differences are 12, 18 and 24; and the third differences are 6 and 6.
Quiz
Which of these is a perfect cube?
What is the units digit of 23³?
Which prime-factor pattern guarantees a perfect cube?
Why can a perfect cube not end in exactly two zeros?
Which equation correctly shows why 1729 is special?
Practice Problems
- Find 12³ and explain the calculation as 12 × 12 × 12.
- Find the cube roots of 27000 and 10648 using suitable reasoning or prime factorisation.
- Find the smallest natural number by which 1323 must be multiplied so that the product is a perfect cube. Also find the cube root of the product.
- Explain why a perfect cube cannot end in exactly five zeros.
- The next taxicab number after 1729 is 4104. Find two different ways to write 4104 as the sum of two positive cubes.
Key Takeaways
• A cube number has the form n³ = n × n × n. • Perfect cubes can end in any digit from 0 to 9. • Cubing triples the number of trailing zeros. • 1729 is the smallest number expressible as the sum of two positive cubes in two different ways. • Groups of consecutive odd numbers can produce perfect cubes. • In prime factorisation, the factors of a perfect cube can be grouped completely into triplets. • Cube root reverses cubing.