Factors
Factors sit at the heart of CAT Number System, and almost every "how many", "what is the sum", or "in how many ways" question about a number traces back to one idea: its prime factorisation. Once you write a number as N = p^a × q^b × r^c, everything about its divisors becomes a counting exercise rather than a listing exercise. You never write out all the factors of 720 — you read them off its prime powers. This chapter builds that fluency: the number-of-factors formula (a+1)(b+1)(c+1), the sum-of-factors product formula, the elegant product-of-factors result N^(d/2), the number of ways to split N into two factors, and the high-yield variations CAT loves — counting only even factors, only odd factors, factors that are perfect squares, or factors divisible by a given number. The same machinery answers "how many divisors are multiples of 6" and "how many ordered pairs (a, b) satisfy a × b = N". Master the prime-factorisation reflex first; every formula here is just a different way of counting what that factorisation already encodes. We keep the India-context CAT/XAT/SNAP framing throughout, favouring the fast structural method over brute listing.
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.
- Always start by writing the prime factorisation N = p^a × q^b × r^c — every divisor question is then pure counting.
- Number of factors = (a+1)(b+1)(c+1). Odd factors: drop the power of 2. Even factors: total − odd.
- Sum of factors σ(N) = product of GP sums (1 + p + ... + p^a) per prime. Sum of reciprocals of factors = σ(N)/N.
- Product of all factors = N^(d/2). To find d from the product: if product = N^k then d = 2k.
- Ways to write N as a product of two factors = d/2, but (d+1)/2 when N is a perfect square (odd d signals a perfect square).
- For "factors divisible by m" with m | N, count the factors of N/m. For "perfect-square factors", use ⌊exponent/2⌋ + 1 per prime.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Forgetting the "+1" in (a+1)(b+1)(c+1) — the exponents themselves do not count the choice of zero copies (which keeps 1 in).
- Adding the prime-power sums instead of multiplying them when computing σ(N).
- Using d/2 for the number of two-factor products when N is a perfect square — it must be (d+1)/2 there.
- Counting even factors as "total minus 1" instead of total minus the odd-factor count.
- Treating the product of factors as N^d instead of N^(d/2) — each pair multiplies to N, and there are only d/2 pairs.
📈 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 Factors 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 divisor formulas
| Prime factorisation | N = p^a × q^b × r^c (p, q, r distinct primes) |
|---|---|
| Number of factors | d(N) = (a+1)(b+1)(c+1) |
| Sum of factors | σ(N) = Π (p^(a+1) − 1)/(p − 1) |
| Product of factors | P(N) = N^(d/2), where d = d(N) |
| Number of odd factors | product of (exponent+1) for odd primes only |
Counting power-tools
| Ways = product of two factors | d/2 if N is not a perfect square; (d+1)/2 if it is |
|---|---|
| Number of even factors | d(N) − d(N with all 2s removed) |
| Factors that are perfect squares | Π (⌊exponent/2⌋ + 1) |
| Factors divisible by m | d(N) counted after dividing N by m (if m | N) |
| Number of co-prime pairs / sum of reciprocals | Σ(1/factor) = σ(N)/N |