Divides
a divides b (a|b) if there exists k ∈ such that b = k·a.
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Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise Arithmetic in IN without losing the thread.
Divides
a divides b (a|b) if there exists k ∈ such that b = k·a.
Euclidean division
For a ∈ , b ∈ : a = b·q + r with 0 ≤ r < b. q = quotient, r = remainder.
GCD(a, b)
The largest integer that divides both a and b. Computed via Euclid's algorithm.
Euclid's algorithm
GCD(a, b) = GCD(b, r) where r is the remainder of a÷b. Repeat until r = 0.
LCM(a, b)
The smallest positive integer that is a multiple of both a and b.
Coprime numbers
a and b are coprime if GCD(a, b) = 1.
Definition
p is prime if p ≥ 2 and its only divisors are 1 and p.
Prime factorization
Every integer n ≥ 2 decomposes uniquely as a product of prime numbers.
Bézout's theorem
a and b coprime ⟺ there exist u, v ∈ such that au + bv = 1.
Theorem 1 - Arithmetic
Application of theorem 1.
Theorem 2 - Arithmetic
Application of theorem 2.
Theorem 3 - Arithmetic
Application of theorem 3.
Theorem 4 - Arithmetic
Application of theorem 4.
Theorem 5 - Arithmetic
Application of theorem 5.
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.