How do you find the least common multiples?

Answer 1

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.

The LCM can be calculated as the product of the two numbers divided by any common factors.

In addition, 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 two given numbers together, and divide by the prime factors they have in common. (Side note: the product of these common prime factors is the two numbers' "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 ÷ 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.

Example: Find the least common multiple of 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).

Example: Find the LCM of 72 and 90 is 360,

72 = 2^3 x 3^2 (8 x 9)

90 = 2 x 3^2 x 5 (2 x 9 x 5)

and 2^3 x 3^2 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.

For two numbers:

First factor each number to its prime factors

Next note the Highest Common Factor

Next multiply your two numbers

Finally divide that figure by the HCF.

EG 24 and 42:

24 = 2 x 2 x 3, 42 = 2 x 3 x 7

Common factors 2 x 3 ie 6

24 x 42 = 1008

1008/6 = 168

The LCM of 24 and 42 is 168.

Answer 2

The way that always works is prime factorization. It is harder with more numbers, though. For example: If your numbers are 6, 8, and 16; Do the prime factorization if you do not know what it is, it is when you take a number and lower it to it's lowest prime numbers. The prime factorization of 6 is 2x3. The prime factorization of 8 is 23. And the prime factorization of 16 is 24. Then comes the hard part. You find which number has the most of a single number. 16 has the most 2s so you keep those and cancel out all of the other ones. You DO NOT use them again. If there is the same number of numbers in two different numbers like if there are two 2s in two different numbers, you can pick which one you want to use. Okay, back to what to do. Now you use the remaining 3. Multiply the four 2s and the 3 to get 48. That is the LCM of 6, 8 , and 16.

I hope this helps.

Answer 3

You cannot.

On the other hand the least common multiple of a set comprising two or more integers is the smallest number which is a multiple of each number in the set. In other words, it is the smallest number which is in the times-table of each of those numbers.

Answer 4

Least Common Multiple of Two NumbersThe LCM (least common multiple) is the smallest positive wholenumber exactly divisible by two or more given whole numbers.Example: the LCM of 14 and 35 is 70 because 70/14=5 and70/35=2, and no number smaller than 70 is exactly divisible by 14and 35.The LCM can be calculated as the product of the two numbers dividedby any common factors.In addition, if one number is a multiple of another number, theirLCM is the smaller number.* Method 1To determine the least common multiple of two numbers, determinethe prime factors of both numbers. Then, determine the primefactors they have in common. Multiply the two given numberstogether, and divide by the prime factors they have in common.(Side note: the product of these common prime factors is the twonumbers' "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 = 60The 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 cancancel out the divisors, so you have as your reduced calculation 5x 2 x 3 x 7 rather than 30 x 42 ÷ 6.* Method 2You can eliminate large multiplications and divisions bydetermining the common factors, finding their product, thenmultiplying by the non-common factors.Example: Find the least common multiple of 125 and 175,125 = 5 x 5 x 5175 = 5 x 5 x 7Common factors = 5 x 5 = 25Uncommon factors = 5 x 7 = 35 and the LCM is 35 x 25 = 875.* Method 3The LCM can also be found for more complex numbers by taking themultiple of the highest power of prime factors from both numbers.(see section below on three numbers).Example: Find the LCM of 72 and 90 is 360,72 = 2^3 x 3^2 (8 x 9)90 = 2 x 3^2 x 5 (2 x 9 x 5)and 2^3 x 3^2 x 5 = 360Least Common Multiple for Three or More Numbers (ExponentialMethod)(*see the related question below)For more than two numbers, LCM can be found as the product of thehighest power of prime factors from all numbers.For example, the LCM of 28, 98, and 350 is 4900 :28 = 22 x 798 = 2 x 72350 = 2 x 52 x 7and (22 x 52 x 72 = 4 x 25 x 49 = 4900).Least Common Multiple of One NumberThere is no "least common multiple" for a single number, becausethe least common multiple is the smallest multiple that two or morenumbers have in common.For two numbers:First factor each number to its prime factorsNext note the Highest Common FactorNext multiply your two numbersFinally divide that figure by the HCF.EG 24 and 42:24 = 2 x 2 x 3, 42 = 2 x 3 x 7Common factors 2 x 3 ie 624 x 42 = 10081008/6 = 168The LCM of 24 and 42 is 168.