Number Systems & Theory
Syllabus Unit: Quantitative Ability (QA)
Why it matters
Chapter overview
Number theory is the silent foundation of CAT Quant — divisibility, LCM/HCF, factor-counting and modular arithmetic rarely appear labelled as such but drive a large share of QA questions, especially remainder-based and factor-based problems where brute force is too slow under exam time pressure.
1Properties of Integers & Divisibility
Divisibility rules let you eliminate wrong answer choices on sight, without ever doing long division.
Learning Objectives
- Recall the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 11 and apply them to multi-digit numbers.
- Classify integers as prime, composite, or neither (1 and 0) and explain why 2 is the only even prime.
- Use the fact that consecutive integers have no common factor other than 1 to simplify counting problems.
- Apply the property that the product of n consecutive integers is always divisible by n!.
Key Terms
CAT QA rewards recognising a number's structure faster than a calculator would. **Divisibility rules** convert a division problem into a quick digit check, which matters enormously under a 40-minute QA section with no calculator.
The rules that come up most
- Divisible by 2: last digit is even (0, 2, 4, 6, 8).
- Divisible by 3: digit sum is divisible by 3.
- Divisible by 4: last two digits form a number divisible by 4.
- Divisible by 9: digit sum is divisible by 9.
- Divisible by 11: (sum of digits at odd positions) − (sum at even positions) is 0 or a multiple of 11.
Prime, composite and co-prime
A **prime** number has exactly two factors; 1 is neither prime nor composite. Two numbers are **co-prime** if their HCF is 1 — this doesn't require either to be prime itself (e.g., HCF(8, 15) = 1).