lcm of 30 and 50
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 30 and 50. To solve this, we will find the multiples of each number and identify the smallest multiple that is common to both.
Answer
$150$
Answer for screen readers
The least common multiple (LCM) of 30 and 50 is $150$.
Steps to Solve
- Find the prime factorization of each number
To find the least common multiple (LCM) of 30 and 50, start by determining their prime factorizations:
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The prime factorization of 30 is: $$ 30 = 2 \times 3 \times 5 $$
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The prime factorization of 50 is: $$ 50 = 2 \times 5^2 $$
- Identify the highest powers of all prime factors
Next, identify each unique prime factor from the factorizations and take the highest power of each prime:
- For prime factor 2, the highest power is $2^1$ (from both).
- For prime factor 3, the highest power is $3^1$ (from 30).
- For prime factor 5, the highest power is $5^2$ (from 50).
- Multiply the highest powers together
Now, multiply the highest powers of each prime factor to find the LCM:
$$ \text{LCM} = 2^1 \times 3^1 \times 5^2 $$
- Calculate the result
Now, calculate the above expression step-by-step:
First, calculate $5^2$: $$ 5^2 = 25 $$
Then multiply: $$ 2 \times 3 \times 25 $$
Calculate this step-by-step:
- $2 \times 3 = 6$
- $6 \times 25 = 150$
Thus, the least common multiple of 30 and 50 is 150.
The least common multiple (LCM) of 30 and 50 is $150$.
More Information
The least common multiple (LCM) is a useful concept in mathematics, especially when working with fractions or finding intervals in real-life applications. The LCM of two numbers is the smallest number that is a multiple of both, making it useful for tasks such as scheduling events or adding fractions with different denominators.
Tips
- Confusing LCM with the greatest common divisor (GCD). Remember that LCM finds the smallest common multiple, while GCD finds the largest common factor.
- Not considering the highest powers of each prime factor, which can lead to an incorrect answer. Always take the highest power for each prime when calculating LCM.
