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How To Find The Least Common Multiple

Asked by Ken Sharp on August 3, 1997:
What is the lowest number which has every number from 1 to 10 as a factor? Is it 2520? This number is divisible by every number between 1 and 10, but is it the lowest number. Does 'the' lowest number have any special name?
This lowest number is often called the least common multiple (or LCM for short). To compute it you must first write out the prime factorization of each of the numbers in question: 2 = 2, 3 = 3, 4 = 2 x 2, 5 = 5, 6 = 2 x 3, 7 = 7, 8 = 2 x 2 x 2, 9 = 3 x 3, and 10 = 2 x 5. Note that we left out 1 as it has no bearing on the value of the LCM.

For each prime listed in one of the above factorizations, find the factorization where it occurs the most. For instance 2 appears most frequently in the factorization of 8, where it appears three times. The most times 3 appears is twice (in 9), and the most 5 and 7 appear is once.

The LCM is the product of all of these primes, each one repeated the number of times it occurs in the factorization where it appears most frequently.

Therefore, the LCM of the numbers from 1 through 10 will be the product of the number 2 (occurring three times), the number 3 (occurring twice), and the numbers 5 and 7 (occuring once). In other words, the LCM is (2^3)(3^2)(5)(7) = 2520.

The reason this is a multiple of every number between 1 and 10 is that any number from 1 to 10 can be written as a product of primes, each occurring at most as often as it does in 2520. Therefore, 2520 is a multiple of that number (it is that number times whatever primes are left over).

For example, 6 = (2)(3), and we can take one 2 and one 3 from 2520 to get 2520 = [(2)(3)][(2^2)(3)(5)(7)] = (6)(420). The same works for any other number from 1 to 10.

The reason that this is the smallest multiple of every number between 1 and 10 is that, in order for one number to divide a second, every prime factor of the first number must divide the second number at least as many times as it divides the first. Thus any common multiple of 1 through 10 must have 2^3 as a factor (since 8 does), must have 3^2 as a factor (since 9 does), and must have 5 and 7 as factors. Therefore any common multiple of 1 through 10 must have (2^3)(3^2)(5)(7) = 2520 as a factor, proving that 2520 is the lowest (or "least") common multiple.

The above arguments also work for finding the LCM of any collection of numbers, not just 1 through 10.

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