Module 3 · Number System
Chapter 31: Prime Numbers
Number System
3 topics
What this chapter covers
The 3 topics below make up Prime Numbers on the CAT. Click any topic to open its notes, examples and practice test.
Study material & tests
📚 Study this chapter
Notes, worked examples and hint-guided practice for every topic.
Start learning →
🎯 Topic tests
8 auto-graded questions per topic, pulled from the question bank.
Practice by topic →
📝 Chapter test
25 fresh mixed-difficulty questions, timed, with full solutions.
Take chapter test →
Formula Reference Sheet
This chapter
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 |
CAT reference
🖩 Graphing Calculator