Find the LCM of 34, 85 and 102.
510
Finding the LCM of 34, 85, and 102
The question asks us to find the Least Common Multiple (LCM) of the numbers 34, 85, and 102.
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. One common method to find the LCM is by using prime factorization.
Steps to Find LCM using Prime Factorization
- Find the prime factorization of each number.
- List all the prime factors that appear in any of the factorizations.
- For each prime factor, take the highest power that appears in any of the factorizations.
- Multiply these highest powers together to get the LCM.
Prime Factorization of the Numbers
Let's find the prime factors for each number:
- For 34: $34 = 2 \times 17$
- For 85: $85 = 5 \times 17$
- For 102: $102 = 2 \times 51 = 2 \times 3 \times 17$
We can summarize the prime factorizations:
| Number | Prime Factorization |
|---|---|
| 34 | $2^1 \times 17^1$ |
| 85 | $5^1 \times 17^1$ |
| 102 | $2^1 \times 3^1 \times 17^1$ |
Calculating the LCM
Now, we list all the prime factors found across the numbers: 2, 3, 5, and 17.
Next, we take the highest power of each prime factor that appears in any factorization:
- Highest power of 2: $2^1$ (from 34 and 102)
- Highest power of 3: $3^1$ (from 102)
- Highest power of 5: $5^1$ (from 85)
- Highest power of 17: $17^1$ (from 34, 85, and 102)
Finally, we multiply these highest powers together to find the LCM:
$\text{LCM}(34, 85, 102) = 2^1 \times 3^1 \times 5^1 \times 17^1$
$\text{LCM}(34, 85, 102) = 2 \times 3 \times 5 \times 17$
$\text{LCM}(34, 85, 102) = 6 \times 5 \times 17$
$\text{LCM}(34, 85, 102) = 30 \times 17$
$\text{LCM}(34, 85, 102) = 510$
Thus, the Least Common Multiple of 34, 85, and 102 is 510.
Revision Table: Understanding LCM Calculation
| Concept | Description | Method Used Here |
|---|---|---|
| LCM | Smallest positive integer divisible by all numbers in a set. | Finding LCM of 34, 85, 102 |
| Prime Factorization | Expressing a number as a product of its prime factors. | Used for 34, 85, and 102 |
| Combining Factors for LCM | Multiplying the highest powers of all prime factors. | $(2^1) \times (3^1) \times (5^1) \times (17^1)$ |
| Result | The calculated LCM value. | 510 |
Additional Information: Importance of LCM
Understanding the LCM is important in various mathematical problems, especially when dealing with fractions to find a common denominator or when solving problems related to cycles or events that repeat at different intervals. The LCM tells us when these different cycles or events will occur simultaneously.
For example, if three events happen every 34, 85, and 102 minutes respectively, they will all happen at the same time again after 510 minutes.
While prime factorization is a fundamental method for finding LCM, other methods like the division method (also known as the ladder method) can also be used, particularly for smaller numbers or when calculating the LCM of more than two numbers simultaneously.
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