least common multiple of 27 and 15
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 27 and 15. To solve this, we need to find the smallest multiple that both numbers share.
Answer
The LCM of 27 and 15 is \( 135 \).
Answer for screen readers
The least common multiple of 27 and 15 is ( 135 ).
Steps to Solve
- Find the prime factorization of each number
To start, we need to break down the numbers 27 and 15 into their prime factors.
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The prime factorization of 27 is: $$ 27 = 3^3 $$
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The prime factorization of 15 is: $$ 15 = 3^1 \times 5^1 $$
- Identify the highest powers of each prime
Next, we look at all the prime factors involved and identify the highest exponent (power) for each prime across both factorizations.
- For the prime factor 3, the highest power between $3^3$ and $3^1$ is $3^3$.
- For the prime factor 5, the highest power is $5^1$ (coming from 15).
- Multiply the highest powers together
Now we multiply these highest powers together to find the least common multiple (LCM).
LCM is calculated as: $$ LCM(27, 15) = 3^3 \times 5^1 $$
- Compute the result
Now we compute the multiplication: $$ LCM(27, 15) = 27 \times 5 = 135 $$
The least common multiple of 27 and 15 is ( 135 ).
More Information
The least common multiple is essential in many areas of math, including fractions and number theory. In this case, knowing the LCM helps in simplifying fractions that share these numbers.
Tips
- A common mistake is to simply multiply the two numbers together without finding the LCM. This can lead to incorrect results, as simply multiplying does not account for the shared multiples.
- Another mistake is to overlook one of the prime factors or to not use the highest exponents when determining the LCM.
