Prime Numbers
Prime numbers are the atoms of arithmetic: every integer above 1 is either a prime or a unique product of primes, and that single fact underpins a surprisingly large slice of CAT Quant. Number-of-factors questions, HCF and LCM, remainder problems, perfect squares, and even the highest power of a number dividing a factorial all trace back to prime factorisation. A prime is a number greater than 1 with exactly two divisors — 1 and itself. So 1 is NOT prime (it has only one divisor), and 2 is the smallest prime and the only even one. CAT does not ask you to recite primes; it tests whether you can factorise fast, count co-primes with Euler’s totient, and reason about divisibility under exam pressure. This chapter builds that toolkit in three layers: primality (including the √N test that lets you check a number quickly), prime factorisation (the engine behind factors, HCF and LCM), and co-primes (where φ(N) counts how many integers below N share no common factor with it). Master these and a whole family of Number System questions becomes mechanical rather than mysterious.
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.
- Primality: only test prime divisors up to √N. If none divide N, it is prime.
- Memorise the 25 primes below 100 — recognising them on sight saves seconds in factor and remainder questions.
- Number of factors of N = product of (exponent + 1). Even factors: fix power of 2 ≥ 1; odd factors: set power of 2 to 0.
- φ(N) = N × ∏(1 − 1/p) over DISTINCT primes — the exponents do not matter for the totient fraction, only the prime list.
- HCF = lowest power of common primes; LCM = highest power of every prime. Always read both off the factorisation.
- Highest power of prime p in N! = ⌊N/p⌋ + ⌊N/p²⌋ + ⌊N/p³⌋ + … (Legendre). Stop when the term becomes 0.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Calling 1 a prime — it has only one divisor, so it is neither prime nor composite.
- Forgetting 2 is prime (and is the only even prime) when listing or counting primes.
- Testing divisibility all the way to N instead of stopping at √N — wastes exam time.
- Using all primes of N (with multiplicity) in φ — only the DISTINCT primes enter the ∏(1 − 1/p) product.
- Assuming co-prime means prime — 8 and 9 are co-prime though neither is prime.
📈 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 Prime Numbers 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
Primality & factorisation
| Primality test | N is prime if no prime ≤ √N divides N |
|---|---|
| Standard form | N = p₁^a × p₂^b × p₃^c × … |
| Number of factors | (a+1)(b+1)(c+1)… |
| Sum of factors | ∏ (p^(e+1) − 1)/(p − 1) |
| Product of factors | N^(d/2), d = number of factors |
Co-primes & totient
| Euler’s totient | φ(N) = N × ∏ (1 − 1/p) |
|---|---|
| φ of a prime | φ(p) = p − 1 |
| φ of prime power | φ(p^k) = p^k − p^(k−1) |
| Co-prime test | a, b co-prime ⇔ HCF(a, b) = 1 |
| Multiplicativity | φ(mn) = φ(m)φ(n) if HCF(m, n) = 1 |