Factorials
A factorial, written n!, is the product of all positive integers up to n: n! = 1 × 2 × 3 × … × n, with the special convention that 0! = 1. The number grows explosively — 10! is already 3,628,800 — so CAT never asks you to compute a big factorial outright. Instead it tests what you can deduce about its structure: how many zeros it ends in, the highest power of a prime or composite that divides it, the rightmost non-zero digit, or how factorials behave inside permutations and combinations. The single tool that unlocks most of these is Legendre’s formula, which counts how many times a prime p divides n! using a clean chain of floor divisions. This chapter builds that skill end to end: trailing zeros (a special case for the prime 5), the general highest-power technique for any prime, the extra care needed for composite divisors, and the basic factorial identities that recur across Number System and Modern Maths. Master factorials and a whole band of seemingly hard CAT questions collapse into two-line arithmetic.
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.
- Trailing zeros of n! = count of 5s only: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … (2s are always in surplus).
- Legendre fast method: keep dividing n by p (taking quotients) and add all the quotients — that is the prime’s exponent.
- Power of a composite: factor it, find each prime’s exponent by Legendre, take the minimum (after dividing by the needed power).
- For 6, 12, 18 etc., the prime 3 is almost always the limiter, not 2 — check the 3-side first.
- For pᵏ (9 = 3², 8 = 2³, 25 = 5²): find the power of p, then floor-divide by k.
- For n ≥ 5, n! ends in 0, so units-digit sums of factorials depend only on 1! through 4! (= 1, 2, 6, 24).
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Counting trailing zeros as just ⌊n/5⌋ and forgetting the ⌊n/25⌋, ⌊n/125⌋ terms (100! has 24 zeros, not 20).
- Stopping Legendre’s sum too early — keep going until pᵏ exceeds n.
- For a composite divisor, taking the power of the larger prime instead of the scarcer one (use the minimum).
- Forgetting to divide by k for prime powers — power of 9 is half the power of 3, not equal to it.
- Assuming 0! = 0; it is defined as 1, which matters in nCr and series formulas.
📈 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 Factorials 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 (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Core definitions & identities
| Factorial | n! = 1 × 2 × 3 × … × n |
|---|---|
| Empty product | 0! = 1! = 1 |
| Recurrence | n! = n × (n − 1)! |
| Ratio of factorials | n! / (n − r)! = n(n−1)…(n−r+1) |
| Combination | ⁿCᵣ = n! / [r!(n − r)!] |
Power & zero-counting tools
| Highest power of prime p in n! (Legendre) | ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … |
|---|---|
| Trailing zeros of n! | ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … |
| Power of composite (e.g. 6 = 2×3) | min of the prime-factor powers in n! |
| Power of pᵏ (e.g. 9 = 3²) | ⌊ (power of p in n!) / k ⌋ |
| Quick count of multiples of m up to n | ⌊n/m⌋ |