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4.3: Greatest Common Divisor and Least Common Multiple Definition: Thegreatest common divisor(GCD) of two natural numbers a and b is the greatest natural number that divides both a and b. Intersection of Sets Method: We write the set of all divisors of each number and then intersect these sets to find the common divisors. The largest element of the intersection is the greatest common divisor. Example: Find the GCD of 12 and 18. Prime Factorization Method: We find the prime factorization of both numbers. The GCD is the product of the common primes, raised to the lowest power that shows up in either prime factorization. Note: If there are no common primes, then the only common factor is 1, so the GCD of these two numbers is 1. In this case, we say that the numbers are relatively prime. Example: Find the GCD of 192 and 360.
Example: Use the Intersection of Sets and Prime Factorization methods to find the GCD of 56 and 84. Definition: Theleast common multiple(LCM) of two natural numbers a and b is the least natural number that is both a multiple of a and a multiple of b. Not Tested: (NumberLine Method/Colored Rods Method) Find the LCM of 3 and 4.
Example: Use the Intersection of Sets and Prime Factorization methods to find the LCM of 12 and 14. Theorem: For any two natural numbers a and b , GCD( a , b ) (^) × LCM( a , b ) = a (^) × b. Proof: See handout on Blackboard. It will appear shortly after class. Example of why this works: Consider the numbers 8 and 12. 8 = 23 = (^23) × 30 12 = (^22) × 31 By considering a 30 , we can take all primes and look at the smallest powers for the GCD and the largest powers for the LCM. GCD(8, 12) = (^22) × 30 LCM(8, 12) = (^23) × 31 So, GCD(8,12) (^) × LCM(8,12) = (^25) × 31 = (2^3 × 30 ) (^) × (2^2 × 31 ) = (^8) × 12. Notice that for each prime, the GCD picks one power and the LCM picks the other power.
