Advanced Arithmetic & Number Theory Cheat Sheet - IIM CAT 2026

1. CAT Exam Pattern & Weightage Analysis

Number System and Arithmetic form the bedrock of the Quantitative Aptitude (QA) section in the IIM Common Admission Test (CAT). Historically, Number Systems accounts for 3 to 5 questions directly, while its foundational concepts (divisibility, prime factorization, and modular remainders) seamlessly penetrate Modern Maths and Algebra topics.

2. Topic 1: Axiomatic Foundations of Numbers & Field Properties

Arithmetic begins with the natural numbers \(\mathbb{N} = \{1, 2, 3, \dots\}\), extended through the integers \(\mathbb{Z}\), rational numbers \(\mathbb{Q}\), and real numbers \(\mathbb{R}\). The fundamental operations of addition (\(+\)) and multiplication (\(\times\)) satisfy the strict field axioms:

• Associativity: \((a + b) + c = a + (b + c)\) and \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
• Commutativity: \(a + b = b + a\) and \(a \cdot b = b \cdot a\)
• Distributivity: \(a \cdot (b + c) = a \cdot b + a \cdot c\)

3. Topic 2: Divisibility Theory & The Euclidean Algorithm

An integer \(a\) is divisible by a non-zero integer \(b\) (denoted \(b \mid a\)) if there exists an integer \(k\) such that \(a = bk\).

The Division Algorithm

For any integers \(a\) and \(b\) with \(b > 0\), there exist unique integers \(q\) (quotient) and \(r\) (remainder) such that:

Computing Greatest Common Divisor (GCD)

The Greatest Common Divisor \(\gcd(a, b)\) can be efficiently computed using the Euclidean Algorithm, relying on the core reduction principle:

4. Topic 3: Fundamental Theorem of Arithmetic & Factorization Trees

Every integer \(n > 1\) can be represented uniquely as a product of prime numbers, up to the order of the factors. Let \(p_1 < p_2 < \dots < p_k\) be distinct primes and \(a_i \ge 1\) be integer exponents:

Canonical Prime Factorization of 60

Consider the prime factorization tree breakdown for \(n = 60\):

• \(60 = 2 \times 30\)
• \(30 = 2 \times 15\)
• \(15 = 3 \times 5\)
• Thus, the canonical prime form is: \[60 = 2^2 \times 3^1 \times 5^1\]

5. Topic 4: Modular Congruences & Remainder Theorems

Let \(m\) be a positive integer called the modulus. We say that \(a\) is congruent to \(b\) modulo \(m\), written as:

This equivalence relation preserves addition, subtraction, and multiplication, serving as the technical foundation for solving complex algebraic remainders in IIM CAT.

6. Frequently Asked Questions (People Also Ask)