Least Common Multiple
- What is the Least Common Multiple?
- How to Find LCM?
- Prime Factorization Method
- Division Method
- Listing Method
- Formula of LCM
- Relationship Between LCM and GCF
- Differences Between LCM and GCF
- Solved Examples
- Practice Problems
- Frequently Asked Question
What is the Least Common Multiple?
Least Common Multiple or, LCM, is a fundamental concept in mathematics. The least common multiple is defined as the smallest multiple that two or more numbers have in common.
For example, if we want to find the LCM of `4` and `8`.
- Multiples of `4: 4, \color{#6F2DBD}8, 12,\color{#6F2DBD}16, 20,\color{#6F2DBD}24, ……`
- Multiples of `8:\color{#6F2DBD}8,\color{#6F2DBD}16,\color{#6F2DBD}24, 32, 40, 48, …….`
So, `8`, `16`, and `24` are common multiples of `4` and `8`. The number `8` is the smallest of the common multiples.
Therefore, `8` is the least common multiple of `4` and `8`.
How to Find LCM?
There are several methods to find LCM:
- Prime Factorization Method
- Division Method
- Listing Method
Prime Factorization Method
Prime factorization is a method used to break down a number into its prime factors, which are the smallest prime numbers that can multiply together to give the original number. This method is essential for finding the Least Common Multiple (LCM) of two or more numbers.
Here are the steps to find the LCM using prime factorization:
`1`. Identify the prime factors of each number: Start by breaking down each number into its prime factors. This involves finding the prime numbers that can be divided evenly into the given number. For example, the prime factors of `12` are `2` and `3` because `12` can be divided by `2` and `3` with no remainder.
`2`. Determine the highest power of each prime factor: Once you have the prime factors of each number, determine the highest power of each prime factor that appears in any of the factorizations. This means identifying the highest exponent for each prime factor across all the numbers. For example, if one number has a prime factor of `2` raised to the power of `3` and another number has a prime factor of `2` raised to the power of `2`, you would choose the higher exponent, which is `3`.
`3`. Multiply the highest powers together: Finally, multiply together the highest powers of each prime factor identified in step `2`. This product will give you the Least Common Multiple (LCM) of the given numbers.
Example: Find the LCM of `15` and `20` using prime factorization.
Solution:
`1`. Identify the prime factors:
Prime factors of `15: 3, 5`
Prime factors of `20: 2, 2, 5`
`2`. Determine the highest power of each prime factor:
For the prime factor `2`, the highest power is `2^2` (from `20`).
For the prime factor `3`, the highest power is `3^1` (from `15`).
For the prime factor `5`, the highest power is `5^1` (from both `15` and `20`).
`3`. Multiply the highest powers together:
LCM`(15, 20) = 2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60`.
So, the LCM of `15` and `20` is `60`.
Division Method
The division method for determining the Least Common Multiple (LCM) is a straightforward technique. It involves dividing the given numbers by their Greatest Common Factor (GCF) and then multiplying the quotients together. Follow these steps to find the LCM using this method:
Step `1`: Identify a prime number that is a factor of at least one of the given numbers. Write this prime number on the left side of the given numbers.
Step `2`: If the prime number identified in Step `1` is a factor of a number, divide that number by the prime and write the quotient below it. If the prime number is not a factor of the number, simply write the number in the row below as it is. Continue these steps until the last row contains `1`.
Example: Let's find the LCM of `12` and `18` using the division method:
Solution:
Step `1`: Start with the smallest prime number `2`, which is a factor of both `12` and `18`.
- Write `2` on the left side of both numbers:
Step `2`: Divide `12` and `18` by `2`.
- Divide `12` by `2` to get `6` and write it below `12`:
- Divide `18` by `2` to get `9` and write it below `18`:
Step `3`: Continue dividing by the largest prime factor that divides at least one of the remaining numbers (`3` in this case).
- Divide `6` and `9` by `3` to get `2` and `3` and so on.
Step `4`: Since there are no more prime factors that divide both numbers, stop.
Step `5`: Multiply the prime factors on the left, considering their highest powers.
- The LCM is the product of these prime factors: \( \text{LCM}(12, 18) = 2 \times 2 \times 3 \times 3 = 36 \)
Therefore, the correct LCM of `12` and `18` is `36`.
Listing Method
The Listing Method, also known as the Common Multiples Method, involves listing the multiples of each number until you find a common multiple.
`1`. List the multiples of each number.
`2`. Identify the smallest common multiple from the lists.
Example: Find the LCM of `8` and `12` using the listing method.
Solution:
`1`. List the multiples of each number:
Multiples of `8: 8, 16,\color{#6F2DBD}24, 32, 40,\color{#6F2DBD}48, …`
Multiples of `12: 12,\color{#6F2DBD}24, 36,\color{#6F2DBD}48, ...`
`2`. Identify the smallest common multiple:
The smallest common multiple of `8` and `12` is `24`.
So, the LCM of `8` and `12` using the listing method is `24`.
Formula of LCM
The Least Common Multiple (LCM) of two numbers \( a \) and \( b \) can also be calculated using the formula:
\( \text{LCM}(a, b) = \frac{ |a \times b| }{ \text{GCF}(a, b) } \)
Where \( \text{GCF}(a, b) \) represents the Greatest Common Factor of \( a \) and \( b \). This formula is derived from the fact that the product of two numbers is equal to the product of their Greatest Common Factor(GCF) and their Least Common Multiple (LCM). So, by rearranging the equation, we can solve for LCM.
Example: Find the LCM of `15` and `20` using the LCM formula given that the GCF of `15` and `20` is `5`.
Solution:
Given that the Greatest Common Factor (GCF) of `15` and `20` is `5`, we can proceed as follows:
The LCM formula states:
\( \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCF}(a, b)} \)
We have: \(a = 15\) and \(b = 20\), and GCF `= 5`. We can substitute these values into the formula:
\( \text{LCM}(15, 20) = \frac{|15 \times 20|}{5} \)
\( \text{LCM}(15, 20) = \frac{300}{5} \)
\( \text{LCM}(15, 20) = 60 \)
So, the LCM of `15` and `20` using the LCM formula, given the GCF is `5`, is `60`.
Relationship Between LCM and GCF
The relationship between the Least Common Multiple (LCM) and the Greatest Common Factor (GCF) of two numbers is defined by a fundamental property in number theory.
When you multiply the LCM and GCF of two numbers, you get the product of the numbers themselves. In mathematical terms:
\( \text{LCM}(a, b) \times \text{GCF}(a, b) = |a \times b| \)
This means that the product of the LCM and GCF of two numbers \( a \) and \( b \) is equal to the absolute value of the product of \( a \) and \( b \).
Differences Between LCM and GCF
Solved Examples
Example `1`: Find the LCM of `18` and `24`.
Solution:
To find the LCM of `18` and `24`, we'll use the prime factorization method:
`1`. Factorize each number:
`18 = 2 × 3^2`
`24 = 2^3 × 3`
`2`. Identify the highest powers of each prime factor:
The highest power of `2` is `2^3`.
The highest power of `3` is `3^2`.
`3`. Multiply these highest powers together to find the LCM:
LCM`(18, 24) = 2^3 × 3^2 = 8 × 9 = 72`
So, the LCM of `18` and `24` is `72`.
Example `2`: Find the LCM of `14`, `21`, and `35`.
Solution:
To find the LCM of `14`, `21`, and `35`, we'll use the prime factorization method:
`1`. Factorize each number:
`14 = 2 × 7`
`21 = 3 × 7`
`35 = 5 × 7`
`2`. Identify the highest powers of each prime factor:
The highest power of `2` is `2^1`.
The highest power of `3` is `3^1`.
The highest power of `5` is `5^1`.
The highest power of `7` is `7^1`.
`3`. Multiply these highest powers together to find the LCM:
LCM`(14, 21, 35) = 2^1 × 3^1 × 5^1 × 7^1 = 2 × 3 × 5 × 7 = 210`
So, the LCM of `14`, `21`, and `35` is `210`.
Practice Problems
Q`1`. Find the LCM of `12` and `30`.
- `15`
- `20`
- `60`
- `90`
Answer: c
Q`2`. What is the least common multiple of `8` and `14`?
- `28`
- `56`
- `112`
- `72`
Answer: b
Q`3`. A group of friends needs to practice marching every `18` days and every `24` days. What is the least number of days needed so they can practice together again?
- `36` days
- `42` days
- `72` days
- `54` days
Answer: c
Q`4`. The length of a rectangular garden is `15` meters and its width is `12` meters. What is the least common multiple of the length and width, representing the smallest area that can be evenly divided into squares?
- `30` meters
- `60` meters
- `60` square meters
- `180` meters
Answer: c
Q5. Two buses arrive at a station every `20` minutes and `30` minutes, respectively. In how many minutes will they arrive at the station again at the same time for the first time?
- `40` minutes
- `60` minutes
- `80` minutes
- `100` minutes
Answer: b
Frequently Asked Questions
Q`1`. What is the Least Common Multiple (LCM)?
Answer: The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of the given numbers.
Q`2`. How do I find the LCM of two numbers?
Answer: You can find the LCM of two numbers using methods such as prime factorization, using multiples, or the division method. These methods involve identifying the prime factors of each number and then determining the highest powers of these factors.
Q`3`. Why is the LCM important?
Answer: The LCM is important in various mathematical applications, including simplifying fractions, adding and subtracting fractions with different denominators, solving equations, and finding common intervals in repeating patterns.
Q`4`. Can the LCM of two numbers be zero?
Answer: No, the LCM of two nonzero numbers is always a positive integer. If one or both of the numbers are zero, the LCM is also zero.
Q`5`. Is the LCM unique for a given pair of numbers?
Answer: Yes, the LCM of a pair of numbers is unique. However, if more than two numbers are involved, there can be multiple common multiples, but the least common one is considered the LCM.