The least common multiple of 24 and 80 is 240.
Understanding the Least Common Multiple (LCM)
The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. It's a fundamental concept in mathematics, particularly useful when adding or subtracting fractions with different denominators, or when solving problems involving cycles and repeating events. For a deeper understanding of LCM, you can explore resources like Wikipedia's page on Least Common Multiple.
Calculating the LCM of 24 and 80
There are several effective methods to determine the LCM of two numbers like 24 and 80. The most common approaches include prime factorization and listing multiples. Both methods consistently show that the LCM of 24 and 80 is 240.
Method 1: Prime Factorization
This method involves breaking down each number into its prime factors.
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Step 1: Prime Factorize each number.
- For 24: $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$
- For 80: $80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5^1$
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Step 2: Identify all unique prime factors and their highest powers.
- The unique prime factors involved are 2, 3, and 5.
- The highest power of 2 is $2^4$ (from 80).
- The highest power of 3 is $3^1$ (from 24).
- The highest power of 5 is $5^1$ (from 80).
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Step 3: Multiply these highest powers together.
- $LCM(24, 80) = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 48 \times 5 = 240$
Method 2: Listing Multiples
This approach involves listing out multiples of each number until a common multiple is found.
| Multiples of 24 | Multiples of 80 |
|---|---|
| 24 | 80 |
| 48 | 160 |
| 72 | 240 |
| 96 | 320 |
| 120 | 400 |
| 144 | |
| 168 | |
| 192 | |
| 216 | |
| 240 |
As demonstrated by both methods, 240 is the least common multiple of 24 and 80.
Practical Applications of LCM
The LCM is not just a theoretical concept; it has practical applications in various fields:
- Fractions: Finding a common denominator when adding or subtracting fractions.
- Scheduling: Determining when events will occur simultaneously (e.g., two buses departing at different intervals).
- Measurement: Calculating the smallest length or quantity that can be divided evenly by different given lengths or quantities.
By understanding how to calculate the LCM, you gain a valuable tool for solving a range of mathematical and real-world problems.
