Perfect Cubes
A perfect cube is any integer that can be written as n³ for some integer n — 1, 8, 27, 64, 125 and so on. They look like a niche topic, but in CAT and the other management entrances they quietly power a whole family of number-system questions: spotting whether a messy number is a cube, extracting cube roots without a calculator, deciding the smallest multiplier that turns a number into a cube, and using the unit-digit pattern to eliminate options in seconds. The reason cubes are exam-friendly is that they carry a unique fingerprint. Unlike squares, where 2 and 8 both end in 4, a cube preserves the last digit almost perfectly — only 2 and 8 swap, and 3 and 7 swap, while every other digit maps to itself. That single fact lets you read the unit digit of a cube root straight off the number. This chapter builds the full toolkit: the properties of cubes and their unit digits, cube roots by grouping digits, the prime-factorisation test (every exponent a multiple of 3), and the beautiful identity that the sum of the first n cubes equals the square of the sum of the first n naturals. Each idea comes with worked CAT-style examples, the fastest mental method, and the traps that cost careless students marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Read the unit digit of a cube root straight off the number: 0,1,4,5,6,9 stay; only 2↔8 and 3↔7 swap.
- Cube root by hand: group digits in threes from the right; last group fixes the unit digit, leading group fixes the rest.
- Perfect-cube test: factorise and check every prime exponent is a multiple of 3. Same test gives the smallest multiplier/divisor.
- Sum of first n cubes = [n(n+1)/2]² — it is always a perfect square, so no cubing is ever needed.
- A partial cube-sum (m+1)³ to n³ = [n(n+1)/2]² − [m(m+1)/2]² (difference of two triangular squares).
- If a + b + c = 0 then a³ + b³ + c³ = 3abc — spot this whenever three terms add to zero.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Treating cube unit digits like square unit digits — forgetting that only 2↔8 and 3↔7 swap while the rest stay put.
- Saying a cube can never end in 2, 3, 7 or 8 (that is squares); cubes can end in any digit 0–9.
- For the smallest cube multiplier, fixing only some exponents — every prime exponent must become a multiple of 3.
- Confusing the sum-of-cubes formula [n(n+1)/2]² with the sum-of-squares formula n(n+1)(2n+1)/6.
- Grouping cube-root digits from the left instead of the right, which throws off the digit count and the unit digit.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Perfect Cubes when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (2 topics) | 2/2 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Cubes & cube roots
| Cube of n | n³ = n × n × n |
|---|---|
| Cube root | ∛(n³) = n |
| Perfect-cube test (prime factors) | every prime exponent is a multiple of 3 |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
Sum-of-cubes identities
| Sum of first n cubes | 1³ + 2³ + … + n³ = [n(n+1)/2]² |
|---|---|
| Link to triangular number | 1³ + 2³ + … + n³ = (1 + 2 + … + n)² |
| Sum of first n cubes | = [Σn]² where Σn = n(n+1)/2 |
| Cube of a binomial | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Identity (a+b+c=0) | a³ + b³ + c³ = 3abc |