Divisibility Summary

Number Theory · Axiom Academy

Let's review the fundamental concepts of divisibility and the foundational principles of number theory. Definition: a | b (read "a divides b") means there exists an integer k such that b = ak Interpretation: b is a multiple of a, or a is a factor of b Key Property: If a | b and b | c , then a | c (transitivity) Zero Rule: Every integer divides 0, but 0 divides only 0 Statement: For any integers a and b (b > 0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b Quotient: q represents the number of times b fits into a Remainder: r is what's left over, always between 0 and b-1 Uniqueness: The quotient and remainder are the only pair satisfying these conditions Essential Divisibility Properties Reflexivity: Every integer divides itself: a | a Transitivity: If a | b and b | c , then a | c Linear Combination: If d | a and d | b , then d | (ax + by) for any integers x, y Antisymmetry: If a | b and b | a , then a = ±b Multiplication: If a | b , then a | bc for any integer c One Divides All: 1 divides every integer; only ±1 divide 1 By 2: Last digit is even (0, 2, 4, 6, 8) By 3: Sum of digits is divisible by 3 By 9: Sum of digits is divisible by 9 Powers of 10: 10 ≡ 1 (mod 3), so 10 n ≡ 1 (mod 3) for all n Digit Sum Trick: A number and its digit sum differ by a multiple of 9 Last Digit Tests: Based on 10 n ≡ 0 modulo 2, 5, or 10 Modular Arithmetic: All tests rely on properties of remainders Base Case: Prove the statement is true for n = 1 (or smallest n)

This is the written version of the interactive lesson above. See the full Number Theory course.