Q&A for How to Find the Least Common Multiple of Two Numbers

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What’s the formula of the least common multiple?

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wikiHow Staff Editor

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The formula is lcm(a, b) = a × b / gcd(a, b), where a and b are the numbers for which you want to find the LCM, and GCD is the greatest common divisor.

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Is there a least common multiple calculator?

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Yes, there are multiple LCM calculators online. Try websites like CalculatorSoup.com or Calculator.net to find calculators for finding the LCM and doing a variety of other common calculations.

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What’s the fastest way to find the least common multiple of two numbers?

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wikiHow Staff Editor

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One quick and easy way to do it is to start by finding the greatest common factor (GCF) of the 2 numbers. Divide the GCF into either one of the 2 numbers, then multiply the result by the other number. This will give you the LCM.

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What is the least common multiple of 105, 210 and 630?

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The least common multiple of 105, 210 and 630 is 630. Both 105 and 210 are factors of 630, therefore 630 is the least common multiple of the three numbers. To verify this, see that 105 x 6 = 630, 210 x 3 = 630, and 630 x 1 = 630.

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How do I calculate the least common multiple and lowest common denominator of fractional numbers?

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To find the lowest common denominator of a fraction, you need to find the LCM of the two denominators.

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What are the two numbers that have the least common multiple of 20?

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Since 20 x 1 = 20, 20 is a multiple of 20. That means 10 and 20 are the two numbers that have an LCM of 20.

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What is the lowest common multiple of 16 and 20?

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16 = 2 x 2 x 2 x 2 20 = 2 x 2 x 5 Since both share a factor of 2 twice, begin by writing 2 x 2. 16 two factors of 2 left over, so the multiplication sentence becomes 2 x 2 x 2 x 2 20 has a factor of 5 left over, so the multiplication sentence becomes 2 x 2 x 2 x 2 x 5 = 80. So, the LCM of 16 and 20 is 80.

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I applied Method 1 to the numbers from Method 4: 210 and 45. I came up with an LCD of 420, which is not right. Where did I go wrong? (210=2x3x5x7, 45=3x3x5) leading to (1x2, 2x3, 1x5, 1x7) leading to 420.

Orangejews

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You are right that the LCM is 2×3×3×5×7. But that comes out to 630, not 420. Perhaps you accidentally wrote a 2 instead of 3 at some point while multiplying out that last step.

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What is the LCM for (a-n) and (a-n)²?

Donagan

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(a-n)².

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Why in the first method is 42 used instead of 2?

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Using 2 instead of 42 in the first method would be too easy to solve. Since 2 is a factor of 20, the first multiple that is common to both of them will be 20. This is because if a number n is a factor of a number p, then the LCM of n and p will always be P. Using the number 42 really shows that we commonly have to solve a problem in which two numbers are not relatively prime, but is relatively large as to pose a challenge to us.

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Why do I need to take prime factors for the least common multiple?

Donagan

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Using prime factors is simply a quicker, more efficient or direct way of looking for an LCM, as shown in Method 2 above.

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If the LCM of two numbers is given, how do I find those two numbers?

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Just by knowing the LCM, you cannot find the exact 2 numbers. You'll have a number of options, though. For example, let the LCM of 2 numbers be 20. The numbers can be 1 and 20, 2 and 20, 4 and 20, 5 and 20, 10 and 20, 4 and 5, 4 and 10, etc. If you have additional constraints (e.g. if the 2 numbers have to add to 9 and have an LCM of 20, then the numbers have to be 4 and 5).

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How do I find the LCM for 17, 24 and 53? There are 2 prime numbers. I can't factor them.

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If you can't find common factors for any of the numbers (other than 1), you just multiply then to get the LCM. LCM of 4 and 5 is 20. 4 and 5 do not have any common factors (other than 1) so 4 * 5 = 20 is their LCM. Similarly, 17, 24 and 53 have no common factors (no common factor for 17 and 24, 17 and 53 or 24 and 53), so their LCM is 17 * 24 * 53 = 21,624.

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So if I am doing method three, what if one number is even and one number is odd?

Donagan

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That just means that they don't share 2 as a factor. They may share some odd number as a factor, however, and you can still use Method 3.

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What's the LCM for 1944 and 324?

Donagan

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The LCM is 1,944. 1,944 = (6)(324).

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How do I find the GCF for a fraction?

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Break the numerator down into its prime factors. Do the same with the denominator. Look at the two groups of prime factors, and find the largest combination of prime factors common to both numerator and denominator. Multiply that group of prime factors together to get the GCF.

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To find the least common multiple, should I factorize by only prime numbers?

Donagan

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Yes. As shown above, whenever you look for a number's factors, you keep breaking the factors down until you're left with prime numbers only. This makes it easier to identify the least common multiples of two (or more) numbers.

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The gcf of 2 different numbers is greater than lcm of the numbers. Always, sometimes, or never?

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Go with 'never'. A factor can never be greater than (any of) the number(s) it's a factor of, and a multiple can never be less than (any of) the number(s) it's a multiple of. So your ordering will be lcm >= larger number > smaller number >= gcf.

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How do I find the LCM of three numbers?

Linnie

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Find the LCM of the first two numbers, and then find the LCM of that result and the last number.

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How do you find the LCM of 12 and 60 without factorization?

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If you notice that 60 is a multiple of 12, it becomes a simple problem: the LCM is 60. If you don't notice that, however, or if that condition does not exist, you'll have to multiply the two numbers together and then reduce the product as much as possible by repeatedly dividing by whatever small prime numbers will divide evenly into the product -- like 3, 5, 7, etc. Reduce until or before you reach the largest of the original numbers (12 and 60 in this case).

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What are the factors of 32?

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1, 2, 4, 8, 16, and 32.

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What if it's a prime number?

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It is the same process. However, if you're trying to find the LCM of two prime numbers, it is the product of the prime numbers.

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