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Answer:
Least Common Multiple of Two Numbers
The LCM (least common multiple) is the smallest positive whole number exactly divisible by two or more given whole numbers. Example: the LCM of 14 and 35 is 70 because 70/14=5 and 70/35=2, and no number smaller than 70 is exactly divisible by 14 and 35. It is the product of the two numbers divided by any common factors. If one number is a multiple of another number, their LCM is the smaller number.
* Method 1
To determine the least common multiple of two numbers, determine the prime factors of both numbers. Then, determine the prime factors they have in common. Multiply the numbers together, and divide by the prime factors they have in common (the product of these is their "greatest common factor").
Example: Find the least common multiple of 12 and 15.
The prime factors of 12 are 2 , 2, and 3 (12 = 2 x 2 x 3)
The prime factors of 15 are 3 and 5 (15 = 3 x 5)
The only prime factor in common is 3.
The least common multiple is (12 x 15) divided by 3. This is 180 / 3 = 60
The least common multiple is 60.
Example: Find the least common multiple of 9 and 11.
The prime factors of 9 are 3 and 3.
The prime factors of 11 are only 11.
There are no prime factors in common.
The least common multiple is 9 x 11 = 99.
Example: Find the least common multiple of 30 and 42.
The prime factors of 30 are 2, 3, and 5.
The prime factors of 42 are 2, 3, and 7.
The prime factors in common are 2 and 3.
The least common multiple is (2 x 3 x 5) x (2 x 3 x 7) / (2 x 3) = 210.
By representing it as a calculation with prime factors, you can cancel out the divisors, so you have as your reduced calculation 5 x 2 x 3 x 7 rather than 30 x 42 รท 6.
* Method 2
You can eliminate large multiplications and divisions by determining the common factors, finding their product, then multiplying by the non-common factors. e.g. for 125 and 175,
125 = 5 x 5 x 5
175 = 5 x 5 x 7
Common factors = 5 x 5 = 25
Uncommon factors = 5 x 7 = 35 and the LCM is 35 x 25 = 875.
* Method 3
The LCM can also be found for more complex numbers by taking the multiple of the highest power of prime factors from both numbers. (see section below on three numbers).
For example, the LCM of 72 and 90 is 360,
72 = 23 x 32 (8 x 9)
90 = 2 x 32 x 5 (2 x 9 x 5)
and 23 x 32 x 5 = 360
Least Common Multiple for Three or More Numbers (Exponential Method)
(*see the related question below)
For more than two numbers, LCM can be found as the product of the highest power of prime factors from all numbers.
For example, the LCM of 28, 98, and 350 is 4900 :
28 = 22 x 7
98 = 2 x 72
350 = 2 x 52 x 7
and (22 x 52 x 72 = 4 x 25 x 49 = 4900).
Least Common Multiple of One Number
There is no "least common multiple" for a single number, because the least common multiple is the smallest multiple that two or more numbers have in common.
The LCM (least common multiple) is the smallest positive whole number exactly divisible by two or more given whole numbers. Example: the LCM of 14 and 35 is 70 because 70/14=5 and 70/35=2, and no number smaller than 70 is exactly divisible by 14 and 35. It is the product of the two numbers divided by any common factors. If one number is a multiple of another number, their LCM is the smaller number.
* Method 1
To determine the least common multiple of two numbers, determine the prime factors of both numbers. Then, determine the prime factors they have in common. Multiply the numbers together, and divide by the prime factors they have in common (the product of these is their "greatest common factor").
Example: Find the least common multiple of 12 and 15.
The prime factors of 12 are 2 , 2, and 3 (12 = 2 x 2 x 3)
The prime factors of 15 are 3 and 5 (15 = 3 x 5)
The only prime factor in common is 3.
The least common multiple is (12 x 15) divided by 3. This is 180 / 3 = 60
The least common multiple is 60.
Example: Find the least common multiple of 9 and 11.
The prime factors of 9 are 3 and 3.
The prime factors of 11 are only 11.
There are no prime factors in common.
The least common multiple is 9 x 11 = 99.
Example: Find the least common multiple of 30 and 42.
The prime factors of 30 are 2, 3, and 5.
The prime factors of 42 are 2, 3, and 7.
The prime factors in common are 2 and 3.
The least common multiple is (2 x 3 x 5) x (2 x 3 x 7) / (2 x 3) = 210.
By representing it as a calculation with prime factors, you can cancel out the divisors, so you have as your reduced calculation 5 x 2 x 3 x 7 rather than 30 x 42 รท 6.
* Method 2
You can eliminate large multiplications and divisions by determining the common factors, finding their product, then multiplying by the non-common factors. e.g. for 125 and 175,
125 = 5 x 5 x 5
175 = 5 x 5 x 7
Common factors = 5 x 5 = 25
Uncommon factors = 5 x 7 = 35 and the LCM is 35 x 25 = 875.
* Method 3
The LCM can also be found for more complex numbers by taking the multiple of the highest power of prime factors from both numbers. (see section below on three numbers).
For example, the LCM of 72 and 90 is 360,
72 = 23 x 32 (8 x 9)
90 = 2 x 32 x 5 (2 x 9 x 5)
and 23 x 32 x 5 = 360
Least Common Multiple for Three or More Numbers (Exponential Method)
(*see the related question below)
For more than two numbers, LCM can be found as the product of the highest power of prime factors from all numbers.
For example, the LCM of 28, 98, and 350 is 4900 :
28 = 22 x 7
98 = 2 x 72
350 = 2 x 52 x 7
and (22 x 52 x 72 = 4 x 25 x 49 = 4900).
Least Common Multiple of One Number
There is no "least common multiple" for a single number, because the least common multiple is the smallest multiple that two or more numbers have in common.
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