Finding the least common multiple (LCM) of two numbers is an important concept in mathematics. The LCM is the smallest positive number that is divisible by both numbers. To find the LCM of 40 and 144, we can use the following steps:
Understanding LCM
The least common multiple is also known as the lowest common multiple or smallest common multiple. The LCM is the smallest positive integer that can be divided evenly by both numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is divisible by both 4 and 6. Other multiples of 4 and 6 such as 24, 36, 48 are not the smallest or least common.
The LCM can be understood as the “common ground” between two numbers – it is the smallest number that both numbers share as a multiple. Finding the LCM is useful for many mathematical operations and problem solving.
Properties of LCM
There are some important properties of LCM that are worth understanding:
- The LCM of two numbers is greater than or equal to the larger of the two numbers
- The LCM of two numbers contains all the prime factors of each number
- The LCM of multiple numbers is the smallest number divisible by all the numbers
- The LCM of two numbers can be found by multiplying the two numbers together, then dividing by their greatest common factor (GCF)
Keeping these properties in mind will help when finding the LCM of numbers.
Finding the LCM of 40 and 144
Now that we understand what LCM is, let’s look at finding the LCM specifically for the numbers 40 and 144:
Step 1: Prime Factorization
The first step is to express each number as a product of prime factors. This allows us to see the unique prime factors in each number.
Prime factorization of 40:
40 = 2 x 2 x 2 x 5
Prime factorization of 144:
144 = 2 x 2 x 2 x 2 x 3 x 3
Step 2: List Unique Prime Factors
Next, we’ll list all the unique prime factors from both numbers:
- 2
- 3
- 5
Step 3: Multiply Unique Prime Factors
To find the LCM, we simply multiply all the unique prime factors together:
LCM = 2 x 2 x 2 x 3 x 5
LCM = 240
Verification
To verify that 240 is indeed the LCM of 40 and 144, we check that it is divisible by both numbers:
- 240 is divisible by 40
- 240 is divisible by 144
Therefore, the LCM of 40 and 144 is 240.
Using the LCM
Finding the LCM has many useful applications. Here are some examples:
Problem Solving
The LCM can be used when solving word problems involving fractions, such as finding the smallest denominator to express fractions with unlike denominators.
Unit Conversions
Converting between units of measurement often involves finding common denominators, which is made easier by using the LCM.
Evenly Spacing Events
The LCM can help determine the smallest interval needed to space events evenly, such as for a medication schedule.
Playing Music
The LCM is useful for determining the shortest loop length needed when musicians play along at different tempos.
Other Examples
Let’s look at finding the LCM for some other number pairs:
LCM of 12 and 18
Prime factors of 12: 2 x 2 x 3
Prime factors of 18: 2 x 3 x 3
Unique prime factors: 2, 3
LCM = 2 x 3 x 3 = 36
LCM of 5 and 7
Prime factors of 5: 5
Prime factors of 7: 7
Unique prime factors: 5, 7
LCM = 5 x 7 = 35
LCM of 15, 20 and 30
Prime factors of 15: 3 x 5
Prime factors of 20: 2 x 2 x 5
Prime factors of 30: 2 x 3 x 5
Unique prime factors: 2, 3, 5
LCM = 2 x 3 x 5 = 30
Finding LCM Using a Table
When dealing with larger numbers, it can be helpful to organize the prime factors in a table. Let’s find the LCM of 420 and 672:
| Number | Prime Factorization |
|---|---|
| 420 | 2 x 2 x 3 x 5 x 7 |
| 672 | 2 x 2 x 2 x 7 x 7 x 7 |
The unique prime factors are 2, 3, 5, and 7. Therefore, the LCM is:
LCM = 2 x 2 x 3 x 5 x 7 = 840
Finding LCM Using Division
There is also a shortcut to find the LCM using division. To use this method:
- Divide the larger number by the smaller number
- If there is no remainder, the smaller number is the LCM
- If there is a remainder, divide the smaller number into the remainder
- Continue dividing until there is no remainder. The last divisor is the LCM.
Let’s use this method to find the LCM of 18 and 24:
24 / 18 = 1 remainder 6
18 / 6 = 3 without remainder
Therefore, the LCM of 18 and 24 is 6.
Finding LCM Using the Formula
There is also a formula that can be used to find the LCM of two numbers:
LCM = (a x b) / GCF
Where a and b are the two numbers, and GCF is their greatest common factor.
Let’s use the formula to find the LCM of 15 and 25:
GCF of 15 and 25 is 5
LCM = (15 x 25) / 5
LCM = 375 / 5
LCM = 75
Conclusion
In summary, here are the key steps to finding the least common multiple of two numbers:
- Factor each number into its prime factors
- List all the unique prime factors
- Take the product of the unique prime factors to get the LCM
The LCM of 40 and 144 is found by factoring into primes, listing the unique factors 2, 3, 5, and multiplying them to get 240. Knowing how to find the LCM is an important mathematical skill with many real-world applications.