LCM of 15 and 27
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 15 and 27. To find the LCM, we need to determine the smallest multiple that is evenly divisible by both of these numbers. This involves finding the prime factors of both numbers and using them to calculate the LCM.
Answer
135
Answer for screen readers
The least common multiple (LCM) of 15 and 27 is 135.
Steps to Solve
- Find the Prime Factorization of Each Number
First, we need to determine the prime factors of 15 and 27.
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The prime factorization of 15 is: $$ 15 = 3 \times 5 $$
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The prime factorization of 27 is: $$ 27 = 3^3 $$
- Identify the Highest Powers of Each Prime Factor
Next, identify the highest power of each prime factor from both factorizations.
- For the prime factor 3, the highest power is $3^3$ (from 27).
- For the prime factor 5, the highest power is $5^1$ (from 15).
- Calculate the Least Common Multiple (LCM)
Now we will calculate the LCM using the highest powers of each prime factor.
The LCM is given by: $$ \text{LCM} = 3^3 \times 5^1 $$
Calculating these values: $$ 3^3 = 27 $$ $$ 5^1 = 5 $$
Now multiply them together: $$ \text{LCM} = 27 \times 5 $$
- Final Calculation
Calculate the final product: $$ \text{LCM} = 135 $$
The least common multiple (LCM) of 15 and 27 is 135.
More Information
The least common multiple is useful in various mathematical applications, including adding and subtracting fractions with different denominators. It’s interesting to note that the LCM is often used in real-life scenarios such as planning events that occur at different frequencies.
Tips
- Not using the highest power: A common mistake is using the lower power of a prime factor instead of the highest.
- Incorrect multiplication: Make sure to multiply correctly, especially with larger numbers. It's easy to make a small arithmetic mistake.
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