How to Calculate LCM of Numbers
Determining the least common multiple, or LCM, of numbers means you are locating the smallest number into which both numbers evenly divide. The numbers 15 and 10 would have a least common multiple of 30, because both numbers evenly divide into it, and there is no smaller multiple where that occurs. An elementary method for determining the least common multiple is to multiply each number by one, two, three, etc. until you can see one that they have in common. This works for small numbers, but this method would be tedious for larger numbers. Using prime factoring works regardless of the size of the numbers. Prime factoring involves locating the prime numbers -- those that only have factors of one and themselves -- that evenly divide into the larger numbers.
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Instructions
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Divide both numbers by two, if it can be done so evenly, and then divide the quotient by two. Keep doing this until the quotient is not evenly divisible by two. Then repeat this procedure by dividing by all the prime numbers, such as three, five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, etc, until you finally come to the last prime number, which will only be divisible by itself.
As an example, find the LCM of 1028: Dividing by two results in 514. Dividing again results in 257. You cannot evenly divide by two anymore, so go to the next prime number. You will notice no prime number evenly divides into it until you reach 257, so you would end there.
As another example, try 1260. Dividing by two equals 630. Dividing by two again gives you 315. Two doesn't go into it any further, so move on to three. Dividing by three equals 105. Dividing by three again gives you 35. Three doesn't evenly divide anymore, but five does. In fact, dividing by five gives you the last prime factor of seven as well.
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List all the prime numbers used for each number, which are called the prime factors. In the examples:
1028 = 2 * 2 * 257
1260 = 2 * 2 * 3 * 3 * 5 * 7
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Count the highest number of any prime factors used by either number.
1028: two 2s, one 257
1260: two 2s, two 3s, one 5, one 7
Since you do not add all the occurrences across both numbers, you would only use a total of two 2s, two 3s, one 5, one 7, and one 257. It might be tempting to use four 2s, but you shouldn't: only use the highest number of them from any one of the numbers.
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Multiply the list of factors you generated to calculate the least common multiple. In the example, you would multiply: two 2s, two 3s, one 5, one 7, and one 257, such that the formula looks like:
LCM = 2 * 2 * 3 * 3 * 5 * 7 * 257
LCM = 323,820
Notice that both numbers evenly divide into this number. Also notice that this number is considerably smaller than you would have achieved by simply multiplying both numbers together, which would have resulted in 1,295,280. If you divided that number by four, you would get the LCM you calculated. That difference is attributed to filtering out those extra two 2s from the prime factors lists.
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