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Least Common Multiple (LCM) Greatest Common Factor (GCF)
A common multiple is a number that is a multiple of two or more numbers. Common multiples of 2 and 3 are 0, 6, 12, 18, ...

The least common multiple (LCM) of two numbers is the smallest number (excluding zero) that is a multiple of both of the numbers.

Here are two least common multiple questions from the Dr. Math archives:

Least Common Multiple Puzzle
What is the smallest number divisible by 1,2,3,4,5,6,8,9,10 that is not 3600?

Running Laps and LCMs
Bill, Bob, and John run 1/3rd, 1/5th, and 1/6th of a lap per minute, respectively. How many laps do they need to run to cross the finish line at the same time?

The least common multiple of two numbers can be found by multiplying one of the numbers by the prime factors of the other number that the two numbers don't have in common.

Here are the prime factors of 40 and 48:

40   2*2*2*5
48   2*2*2*2*3

We can multiply 48 by 5 (the only prime factor of 40 not shared by 48), or we can multiply 40 by 2*3. Either way, we'll get 240.

What about the LCM of the numbers 16 and 24? Factor:

16   2*2*2*2
24   2*2*2*3

We can multiply 16 by 3 (the only prime factor of 24 not shared by 16), to find the LCM: 3 * 16 = 48.
The greatest common factor (GCF) is the greatest factor that is common to two or more numbers (they share it).

The greatest common factor of two (or more) numbers is the product of all the prime factors the numbers have in common.

If you want to find the greatest common factor of 16 and 24, express both as products of their prime factors, and look for factors common to both:

16   2*2*2*2
24   2*2*2*3

There are three 2's common to both numbers, so 2*2*2 = 8 is the "greatest common factor" (GCF) of 16 and 24.

Here are two step-by-step explanations from the Dr. Math archives:

Finding the Greatest Common Factor of Two Different Numbers
Finding the GCF of two different numbers after using a factoring tree to find their factors.

Finding LCDs, LCMs, and GCFs
How do you find the greatest common factor, the least common multiple, and the least common denominator?

For more about factoring, see:
Prime Factoring.

Return to
Elementary Multiplication
Middle School Factoring Numbers
Index of Selected Answers to Common Questions

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