lcm of 27 and 15
Understand the Problem
The question is asking us to find the least common multiple (LCM) of the numbers 27 and 15. To solve this, we will consider the prime factors of each number and determine the smallest multiple that both numbers share.
Answer
$135$
Answer for screen readers
The least common multiple (LCM) of 27 and 15 is $135$.
Steps to Solve
- Find the Prime Factors of Each Number
First, we will find the prime factors of both numbers.
For 27:
- The prime factorization is $27 = 3^3$.
For 15:
- The prime factorization is $15 = 3^1 \cdot 5^1$.
- Identify the Highest Power of Each Prime Factor
Next, we will take the highest power of each prime factor from both numbers:
- 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).
- Multiply the Highest Powers Together
Now we will multiply these highest powers together to find the LCM:
$$ \text{LCM} = 3^3 \cdot 5^1 $$
Calculating this gives:
$$ \text{LCM} = 27 \cdot 5 = 135 $$
The least common multiple (LCM) of 27 and 15 is $135$.
More Information
The least common multiple is the smallest number that is a multiple of both numbers. In this case, $135$ is the first number that both $27$ and $15$ can divide into without leaving a remainder. LCMs are useful in problems involving fractions and finding common denominators.
Tips
- Forgetting to Consider All Prime Factors: When calculating the LCM, make sure to take the highest power of every prime factor from both numbers rather than just the ones present in one of them.
- Miscalculating the Multiplication: Double-check calculations when multiplying the highest powers to avoid arithmetic errors.
