Number system

Number System – Advanced Study Notes

Target Exam: IBPS Clerk – Quantitative Aptitude
Level: MIT‑style, first‑principles with rigorous reasoning & edge‑case analysis

Why these notes?
The “Number System” section of IBPS is often the crux of quantitative aptitude. A solid grasp of base‑representation, divisibility, and modular arithmetic not only scores well in the exam but also lays a foundation for higher mathematics.

Structure

1. Foundations – base‑b representation, conversions, uniqueness.
2. Rational/irrational expansions – finite vs repeating, period theory.
3. Divisibility – classical rules and generalizations.
4. Prime‑number theory – Euclid, Wilson, Fermat.
5. GCD, LCM, modular arithmetic, Euler’s theorem, CRT.
6. Edge cases & pitfalls.
7. Advanced topics & applications.
8. Practice problems (simple → advanced).
9. One challenging problem with full solution.

Notation
b – any integer base (b ≥ 2).
n – positive integer.
gcd(a,b) – greatest common divisor.
φ(n) – Euler’s totient function.
ord_b(n) – multiplicative order of b modulo n (smallest k > 0 s.t. b^k ≡ 1 (mod n)).

1. Foundations – Base‑b Representation

1.1 Decimal, Binary, Octal, Hexadecimal

• Decimal: base 10, digits 0–9.
• Binary: base 2, digits 0,1.
• Octal: base 8, digits 0–7.
• Hexadecimal: base 16, digits 0–9, A–F.

1.2 Representation in Base b

Definition
Any integer N (positive or negative) can be written uniquely as
[ N = \sum_{i=0}^{k} d_i,b^i,\quad d_i\in{0,1,\dots,b-1}, ] where k = ⌊log_b |N|⌋ for N > 0.
For negative numbers, use a minus sign or two’s complement in binary contexts.

Theorem (Uniqueness)
For a fixed base b and fixed digit set, the representation above is unique.

Proof Sketch
Suppose two distinct expansions exist. Subtracting yields a non‑zero integer that is divisible by b but less than b, impossible unless the difference is zero. ∎

1.3 Converting Between Bases