I find the concept of the "double bubble" to be helpful for many GCF/LCM problems. Consider, for example, the numbers 6 and 15. 6 = 2*3 15 = 3*5 Their greatest common factor (GCF) is 3, because it is their only common factor greater than 1. The least common multiple (LCM) can be written as the product of the factors of these numbers like this LCM = (2[3)5] = 30 The numbers in the () parentheses (bubble) are the factors of 6. The numbers in the [ ] brackets (bubble) are the factors of 15. The number in the overlapping area [ ) is the greatest common factor (GCF) of the two numbers. We can also see that the LCM of two numbers is their product divided by their GCF. LCM = (2*3)*(3*5)/3 = 2*3*5
The greatest common factor (GCF) of two numbers can be found reliably using the Euclidean Algorithm. Divide the largest by the smallest. If the remainder is zero, the smallest is the GCF. If the remainder is not zero, use the smallest and the remainder as the two numbers, and repeat. Example GCF(6, 15) = ? 15/6 = 2 remainder 3 The smallest number and the remainder are the two numbers we use for the next go-around. 6/3 = 2 remainder 0. The remainder is 0, so 3 is the GCF. (Which we already knew.) If you think about it a bit, you realize the GCF of two numbers will never be larger than their difference. Example: 12 and 14 have a difference of 2, so 2 is the largest the GCF could be. As it happens, their GCF is 2. (If you use the Euclidean Algorithm, you try 14/12 = 1 remainder 2, and 12/2 = 6 remainder 0. Thus 2 is the GCF. Of course the remainder of 2 is just the difference of the two numbers 14 and 12.) If you are working with the GCF or LCM of more than 2 numbers, find the GCF/LCM of two of them, then use that as one of the numbers and find the GCF/LCM of that and the next number on your list. Example LCM of 10, 15, 35 = LCM(LCM(10, 15), 35) = LCM(30, 35) We know 35-30=5 is the GCF, because both are divisible by 5. Thus, the LCM is 30*35/5 = 30*7 = 210
The greatest common factor (GCF) of two numbers can be found reliably using the Euclidean Algorithm. Divide the largest by the smallest. If the remainder is zero, the smallest is the GCF. If the remainder is not zero, use the smallest and the remainder as the two numbers, and repeat. Example GCF(6, 15) = ? 15/6 = 2 remainder 3 The smallest number and the remainder are the two numbers we use for the next go-around. 6/3 = 2 remainder 0. The remainder is 0, so 3 is the GCF. (Which we already knew.) If you think about it a bit, you realize the GCF of two numbers will never be larger than their difference. Example: 12 and 14 have a difference of 2, so 2 is the largest the GCF could be. As it happens, their GCF is 2. (If you use the Euclidean Algorithm, you try 14/12 = 1 remainder 2, and 12/2 = 6 remainder 0. Thus 2 is the GCF. Of course the remainder of 2 is just the difference of the two numbers 14 and 12.) If you are working with the GCF or LCM of more than 2 numbers, find the GCF/LCM of two of them, then use that as one of the numbers and find the GCF/LCM of that and the next number on your list. Example LCM of 10, 15, 35 = LCM(LCM(10, 15), 35) = LCM(30, 35) We know 35-30=5 is the GCF, because both are divisible by 5. Thus, the LCM is 30*35/5 = 30*7 = 210
