The least common multiple (LCM) is the smallest number that is divisible into other numbers. This means that when multiplying multiple numbers we want to find the smallest number that they both go into. The mathematical way we would represent this is LCM(a,b); where a and b are both constants(numbers). The purpose of finding the LCM is for computations with fractions so let’s look at some fractional problems and find its LCM.
1/3+1/2 = 2(1/3)+3(1/2)= 2/6+3/6=(2+3)/6=5/6
When we add these fractions we want to simplify the denominator (bottom of the fraction) into its factors and then multiply each fraction (top and bottom) by the factor it is missing. Once we do this we have our LCM as our denominator and we have a common base which allows us to add our fractions together.
1/6+1/8=1/(3*2)+1/(2*2*2) = (2*2)(1/6)+(3)(1/8)=4/24+3/24 = (4+3)/24=7/24
When we are adding these fractions we used the same process as the first problem but since we did not already have our numbers simplified in factored form we needed to separate the numbers and multiply by what we missing. When we look at the 1/6 we see its factors are 2,3; and the 1/8 has 2,2,2. This means that our LCM is 2*2*2*3=24 because we need each fraction to have the same base so we multiplied the two 2’s into the 1/6 and the 3 into the 1/8. A proof that this works is:
1/6+1/8 = 8*(1/6)+6*(1/8)= 8/48+6/48=14/48= (7*2)/(24*2)=7/24
As we notice if we just find a common base and add our fractions we get a non simplified result. Then if we simplify the result we can simplify it by the factor that was duplicated in our factored form. In other words our factors (2*3) , (2*2*2) both have a two in them so instead of simplifying our problem we had our extra 2. So instead of having 2*2(2*3), 3*(2*2*2) we had (2*3*2*2*2) which gave us an extra two.
2/3+5/7 = 7(2/3)+3(5/7) = 14/21+15/21 = (14+15)/21=29/21= 21/21+8/21= 1+8/21
In this example we notice our numerator (top part of the fraction) is not 1. This does not make any difference toward finding our LCM but we just need to keep in mind that we have to multiply our original numerator by our LCM to get our new numerator over our common base.
In summary, a LCM is just a way to simplify your work with fractions so when you add or subtract them you will not need to later simplify. To find your LCM simplify each fractions denominator into its factors; then remove any duplicates that both fractions have. Then you multiply every remaining factor to get your common base and multiply each fractions numerator by the factors that were missing in its original form. Then you create one big fraction and do your operations in the numerator to find your final answer.
