Least Common Multiple

Number Theory · Axiom Academy

Understanding the smallest shared multiple and its applications in number theory Definition: The least common multiple of two positive integers a and b , denoted lcm( a , b ), is the smallest positive integer that is divisible by both a and b . In other words, lcm( a , b ) is the smallest number that appears in both the list of multiples of a and the list of multiples of b . 2. Finding LCM Using Prime Factorization The most efficient way to find the LCM is using prime factorization. The key principle is: Take max exponents: 2 max(2,1) × 3 max(1,2) = 2 2 × 3 2 = 4 × 9 = 36 The least common multiple satisfies several important algebraic properties: Commutative: lcm( a , b ) = lcm( b , a ) Associative: lcm(lcm( a , b ), c ) = lcm( a , lcm( b , c )) Divisibility: If a divides b , then lcm( a , b ) = b The concept of LCM appears throughout mathematics and real-world problem solving:

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