A number k is a common factor of two numbers m and n if it is a factor of each of the numbers.
For example, 5 is a common factor of 10 and 15 because 5 × 2 = 10 and 5 × 3 = 15.
A number m is a common multiple of two numbers n and k if it is a multiple of each of the numbers.
For example, 18 is a common multiple of 6 and 9 because 6 × 3 = 18 and 9 × 2 = 18.
The smallest number that is a common multiple of two (or more) numbers is called the least common multiple (LCM).
To find the LCM of given numbers, write each given number as a product of primes. Next, delete the common factors (if any) from all but one of the products. Finally, multiply the remaining factors to obtain the LCM.
For example to find the LCM of 18, 45 and 81, each number is expressed as 18 = 2 × 3 × 3, 45 = 3 × 3 × 5 and 81 = 3 × 3 × 3 × 3. Next, 3 × 3 = 9 is the common factor; so, delete it in all but one of the products. Finally, the LCM is 2 × 3 × 3 × 3 × 3 × 5 = 810.
Note that the LCM is the product of the highest power (i.e., number of times a factor occurs) of each separate factor.
Example
Professor Morton grades a test paper in 14 minutes, whereas Professor Newstein takes only 12 minutes. If both professors start grading at 10:00 a.m., when will it be the first time they will finish grading a test paper at the same time? Solution.
Professor Morton will grade m papers in 14m minutes.
Professor Newstein will grade n papers in 12n minutes.
To finish grading at the same time, 14m = 12n.
Note that m and n must be integers because they denote number of test papers.
Thus, the task is to find a common multiple of 14 and 12.
Since the first time they finish together is needed, the LCM is required.
Now, 14 = 2 × 7 and 12 = 2 × 2 × 3. When the common factor 2 is deleted in one of the products, the LCM is 2 × 2 × 3 × 7 = 84.
The two professors will finish grading a test paper at the same time 84 minutes after starting, i.e., at 11:24 a.m.
