What is the 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 find the LCM, we will factor both numbers into their prime factors and then use these factors to calculate the lowest common multiple.
Answer
The least common multiple of 30 and 50 is $150$.
Answer for screen readers
The least common multiple (LCM) of 30 and 50 is 150.
Steps to Solve
- Factor the numbers into prime factors
We start by determining the prime factorization of both numbers.
For 30:
$$ 30 = 2 \times 3 \times 5 $$
For 50:
$$ 50 = 2 \times 5^2 $$
- Identify the highest powers of all prime factors
Next, we will identify the highest power of each prime number present in both factorizations.
- The prime factor 2 appears as $2^1$ in both.
- The prime factor 3 appears as $3^1$ (only in 30).
- The prime factor 5 appears as $5^2$ (from 50).
- Calculate the LCM
To find the LCM, we multiply the highest powers of all the prime factors together:
$$ LCM = 2^1 \times 3^1 \times 5^2 $$
Calculating that yields:
$$ LCM = 2 \times 3 \times 25 $$
$$ = 150 $$
The least common multiple (LCM) of 30 and 50 is 150.
More Information
The least common multiple is useful in finding common denominators in fractions, and it helps in solving problems involving synchronous events like finding out when two cycles will coincide.
Tips
- Forgetting to include a prime factor when identifying the highest powers.
- Confusing the LCM with the greatest common divisor (GCD).
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