What is Least Common Multiple?
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. In other words, it’s the smallest number that can be divided evenly by each of the numbers without leaving a remainder.
For example, the LCM of 6 and 9 is 18, because 18 is the smallest multiple that is common to both 6 and 9.
How to Find the LCM?
There are three methods to find the LCM of two or more numbers:
- List the Multiples Method
- Prime Factorization Method
- Division Method
List the Multiples Method
In this method, we list the multiples of each number and find the smallest one that appears in both lists.
Example 1:
Find the LCM of 4 and 6
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28…
- Multiples of 6: 6, 12, 18, 24, 30…
- The common multiples are 12 and 24, but the smallest one is 12. So, the LCM of 4 and 6 is 12.
Example 2:
Find the LCM of 3, 4, and 5 by listing their multiples.
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39…
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40…
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40…
The smallest number that appears in the multiples of 3, 4, and 5 is 60. So, the LCM of 3, 4, and 5 is 60.
Prime Factorization Method
This method involves breaking down each number into its prime factors and then using those factors to find the LCM.
Example 1:
Find the LCM of 8 and 12
- Prime factors of 8: 2 × 2 × 2
- Prime factors of 12: 2 × 2 × 3
To find the LCM, take the highest powers of all prime factors:
- LCM = 2 × 2 × 2 × 3 = 24
So, the LCM of 8 and 12 is 24.
Example 2:
Find the LCM of 12, 16 and 18.
- Prime factors of 12: 2 × 2 × 3 = 2²x3¹
- Prime factors of 16: 2 x 2 x 2 × 2 = 2⁴
- Prime factors of 18: 2 x 3 x 3 = 2¹ x 3²
Identify the highest powers of each prime.
The highest power of 2 is 2⁴.
The highest power of 3 is 3².
Multiply all the highest powers.
LCM (12, 16, 18) = 2⁴ x 3²
LCM (12, 16, 18) = 16 x 9 = 144
Division Method
This method involves breaking down each number into its prime factors and then using those factors to find the LCM.
Steps:
Write the Numbers in a Row: Start by writing the numbers you want to find the LCM for, all in one row.
Divide by the Smallest Prime Number: Begin dividing all the numbers by the smallest prime number (starting with 2). If the number is divisible, divide it. If not, leave it as is.
Continue Dividing: Keep dividing the results by prime numbers (2, 3, 5, 7, etc.) until you can no longer divide any of the numbers by a prime number.
Multiply the Divisors: Multiply all the prime numbers used in the division process. The result is the LCM.
Example 1:
Find the LCM of 12 and 15 using Division method.
- Write the Numbers.
12, 15 - Both 12 and 15 are divisible by 3. Divide them by 3.
4, 5 - There are no more common factors. 4 can be divided by 2.
2, 5 - 2 is again divided by 2.
1, 5 - 5 is again divided by 5
1, 1 - The divisors that have been used so far are 3, 2, 2, 5. Multiply them to get the LCM.
3x2x2x5 = 60
So, the LCM (12, 15) = 60.
Real-Life Applications of LCM
The Least Common Multiple (LCM) is not just a math concept but also something we use in everyday life, often without realizing it! Let’s explore some practical applications of LCM:
Scheduling Events
One of the most common uses of LCM is in scheduling events that happen at regular intervals. For example:
- If two buses arrive at a bus stop every 2 hours and every 3 hours, the LCM helps find when they will arrive together.
- By finding the LCM of 2 and 3 (which is 6), you can determine that both buses will arrive together every 6 hours.
Synchronizing Cycles
LCM is used when we need to synchronize cycles or repeated events that happen after different intervals of time.
- Example: If a light flashes every 6 seconds, another light flashes every 8 seconds, and you want to know when they will flash together, you can use the LCM of 6 and 8. The LCM is 24, so the lights will flash together every 24 seconds.
Finding Common Time Frames
In classrooms, factories, or offices, LCM can help find when activities or processes that operate on different schedules will coincide.
- Example: In a classroom, if one student has a break every 5 days and another every 7 days, the LCM of 5 and 7 (which is 35) will tell you that both students will have a break on the same day every 35 days.
Working with Fractions
LCM is essential when adding or subtracting fractions with different denominators. The LCM of the denominators gives you the least common denominator (LCD), which allows you to combine the fractions easily.
- Example: If you want to add 1/4 and 1/6, you first need to find the LCM of 4 and 6, which is 12. Then, you convert the fractions to have the same denominator and add them.
Tiling Floors or Designing Patterns
LCM can be helpful when you’re tiling a floor or designing patterns with tiles of different sizes. The LCM helps you determine how to arrange the tiles so they fit together without cutting any tiles.
- Example: If one tile measures 4 inches and another measures 6 inches, the LCM of 4 and 6 (which is 12) tells you that the tiles will align every 12 inches.
Rotational or Mechanical Gears
LCM is used in mechanical systems with gears or wheels that rotate at different speeds. It helps find when the gears will align or return to their starting positions.
- Example: If one gear rotates every 5 seconds and another rotates every 8 seconds, the LCM of 5 and 8 is 40. This means the gears will align every 40 seconds.
Musical Patterns
LCM is used in music to find when repeating patterns of notes or beats will align. Musicians use it to create rhythmic patterns that repeat at regular intervals.
- Example: If one beat pattern repeats every 3 seconds and another every 4 seconds, the LCM of 3 and 4 (which is 12) tells you that both patterns will align every 12 seconds.
LCM Practice Questions
- Find the LCM of 30 and 40 by listing the multiples method.
- Find the LCM of 10, 15 and 30 using the Division Method.
- If one machine completes a task every 10 seconds and another every 6 seconds, how often will they complete the task at the same time?