What is the Smallest Multiple of 12 and 6?

Understanding the Concept

The least common multiple (LCM), also known as the lowest common multiple or least common divisor, is a fundamental concept in mathematics. It refers to the smallest positive integer that can be divided evenly by two given numbers – a and b. For instance, the LCM of 2 and 3 is 6, while the LCM of 6 and 10 is 30. In other words, the LCM of multiple numbers is the smallest number that is divisible by all the numbers in the set.

Using a Calculator

To find the LCM of a set of numbers, you can utilize a helpful calculator that not only provides the result but also demonstrates the step-by-step process.

Start by inputting the numbers you want to find the LCM for. You can separate the numbers using commas or spaces, but avoid using commas within the numbers themselves. For example, input “2500, 1000” instead of “2,500, 1,000.”

Different Methods to Find the LCM

There are several methods you can employ to determine the LCM. Here are six commonly used approaches:

Method 1: Listing Multiples

To find the LCM by listing multiples, follow these steps:

1. List the multiples of each number until you find at least one multiple that appears in every list.
2. Identify the smallest number that appears in all the lists – this number is the LCM.

For example, let’s find the LCM of 6, 7, and 21:

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
• Multiples of 7: 7, 14, 21, 28, 35, 42, 56, 63
• Multiples of 21: 21, 42, 63

Since 42 is the smallest number that appears in all the lists, the LCM of 6, 7, and 21 is 42.

Method 2: Prime Factorization

To find the LCM using prime factorization, follow these steps:

1. Determine the prime factors of each given number.
2. List all the prime numbers found, repeating them as many times as they occur most frequently in any one number.
3. Multiply the list of prime factors together to find the LCM.

For example, let’s find the LCM of 12 and 30:

• Prime factorization of 12: 2 × 2 × 3
• Prime factorization of 30: 2 × 3 × 5

By multiplying all the prime factors together, we get: 2 × 2 × 3 × 5 = 60. Thus, the LCM of 12 and 30 is 60.

Method 3: Prime Factorization using Exponents

This method is similar to prime factorization but incorporates exponents. Here are the steps:

1. Determine the prime factors of each given number and write them in exponent form.
2. List all the prime numbers, using the highest exponent found for each.
3. Multiply the list of prime factors with exponents together to find the LCM.

For example, let’s find the LCM of 12, 18, and 30:

• Prime factors of 12: 2 × 2 × 3 = 2^2 × 3^1
• Prime factors of 18: 2 × 3 × 3 = 2^1 × 3^2
• Prime factors of 30: 2 × 3 × 5 = 2^1 × 3^1 × 5^1

By multiplying all the prime factors with the highest powers, we get: 2^2 × 3^2 × 5^1 = 180. Thus, the LCM of 12, 18, and 30 is 180.

Method 4: Cake/Ladder Method

The cake method, also known as the ladder method, is a simple division-based approach to finding the LCM. Follow these steps:

1. Write down your numbers as a row, resembling a cake layer.
2. Divide the numbers in the row by a prime number that evenly divides into two or more of them, and carry the result down to the next layer.
3. If any number in the layer is not evenly divisible, write it down as it is.
4. Continue dividing the cake layers by prime numbers until no more primes can evenly divide two or more numbers.
5. The LCM is the product of the numbers in the L shape formed by the left column and bottom row, excluding 1.

For example, let’s find the LCM of 10, 12, 15, and 75:

• The LCM is 2 × 3 × 5 × 2 × 5 = 300. Therefore, the LCM of 10, 12, 15, and 75 is 300.

Method 5: Division Method

The division method is another effective way to find the LCM. Follow these steps:

1. Arrange the given numbers in a row at the top of a table.
2. Starting with the lowest prime numbers, divide the row of numbers by a prime number that evenly divides at least one of the numbers, and bring down the results to the next row.
3. If any number in the row is not evenly divisible, write it down as it is.
4. Continue dividing the rows by prime numbers that divide evenly into at least one number.
5. Once the last row of results consists entirely of 1’s, stop the process.
6. The LCM is the product of the prime numbers in the first column.

For example, let’s find the LCM of 10, 18, and 25:

• The LCM is 2 × 3 × 3 × 5 × 5 = 450. Therefore, the LCM of 10, 18, and 25 is 450.

Method 6: Using the Greatest Common Factor (GCF)

The LCM can also be found using the GCF of a set of numbers. Employ the following formula:

LCM(a, b) = (a × b) / GCF(a, b)

For example, let’s find the LCM of 6 and 10:

• Find the GCF of 6 and 10: GCF(6, 10) = 2
• Use the formula: (6 × 10) / 2 = 60 / 2 = 30
• Therefore, the LCM of 6 and 10 is 30.

Additional Insights

The LCM possesses certain properties:

• The LCM is associative: LCM(a, b) = LCM(b, a)
• The LCM is commutative: LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c))
• The LCM is distributive: LCM(da, db, dc) = d × LCM(a, b, c)
• The LCM is related to the GCF: LCM(a, b) = (a × b) / GCF(a, b) and GCF(a, b) = (a × b) / LCM(a, b)

These properties assist in mathematical calculations involving the LCM.

References:

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition, New York, NY: CRC Press, 2003 p. 101.

[2] Weisstein, Eric W. Least Common Multiple. From MathWorld – A Wolfram Web Resource.