LCM of 3, 5, and 11
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 3, 5, and 11. To find the LCM, we identify the smallest positive integer that is a multiple of each of the given numbers.
Answer
$165$
Answer for screen readers
The least common multiple (LCM) of 3, 5, and 11 is $165$.
Steps to Solve
- List the Multiples First, we can list some multiples of each number:
- Multiples of 3: $3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...$
- Multiples of 5: $5, 10, 15, 20, 25, 30, ...$
- Multiples of 11: $11, 22, 33, 44, 55, 66, 77, 88, 99, 110, ...$
- Identify Common Multiples Next, we need to find the common multiples from the lists above.
From our lists, we can see that both 3 and 5 share some common multiples, and we can check them against multiples of 11:
- Common multiples of 3 and 5: $15, 30, ...$
- Check for the Least Common Multiple Finally, we need to check these common multiples against the multiples of 11. The smallest common multiple we found was $15$, but it is not in the multiples of 11. Next, we try $30$.
- Check if $30$ is a multiple of $11$: $$ 30 \div 11 \approx 2.727 \quad \text{(not a whole number)} $$
- Thus, $30$ is not a common multiple.
Next, we can continue our search; we find that $3 × 5 × 11 = 165$.
- Conclusion The least common multiple of 3, 5, and 11 is therefore $165$.
The least common multiple (LCM) of 3, 5, and 11 is $165$.
More Information
The least common multiple is particularly useful in problems that involve adding or subtracting fractions, or when trying to find a common time frame for events that occur at different intervals. The LCM helps to simplify computations and allows for easier analysis.
Tips
- One common mistake is thinking the LCM is simply the product of the numbers. In this case, while the product ($3 \times 5 \times 11 = 165$) happens to be the LCM, that's not a general rule that applies to all sets of numbers, especially if they share factors.
