Divisibility Definition and Properties

Number Theory · Axiom Academy

LESSON Divisibility Definition and Properties Understanding the formal definition and fundamental properties of divisibility in number theory 1. Formal Definition of Divisibility Definition: Let a and b be integers with a ≠ 0. We say that " a divides b " if there exists an integer k such that b = ak . Notation: We write a | b to denote " a divides b ". Read as: " a divides b ", " a is a divisor of b ", or " b is divisible by a ". The key insight is that divisibility requires the existence of an integer multiplier k . This animation demonstrates how 3 | 12 because we can find an integer k = 4 such that 12 = 3 × 4. 2. Reflexivity and Transitivity Reflexivity: For any non-zero integer a , we have a | a . This is because a = a × 1, and 1 is an integer. Transitivity: If a | b and b | c , then a | c . This property allows us to chain divisibility relationships. The animation below illustrates transitivity by showing how divisibility flows through a chain of numbers. 3. Linear Combinations Property Theorem: If a | b and a | c , then a | ( bx + cy ) for any integers x and y . Proof: Since a | b , we have b = ak₁ for some integer k₁ . Since a | c , we have c = ak₂ for some integer k₂ . Therefore: Since k₁x + k₂y is an integer (sum and product of integers are integers), this shows that a | ( bx + cy ). Understanding what divisibility is and what it is not is crucial. (-3) | 15 because 15 = (-3) × (-5) 1 | n for any integer n because n = 1 × n

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