Least common multiple of 30 and 75
Understand the Problem
The question is asking for the least common multiple (LCM) of the two numbers 30 and 75. To solve this, we typically find the prime factorization of both numbers, then use those factors to determine the LCM.
Answer
The LCM of 30 and 75 is $150$.
Answer for screen readers
The least common multiple (LCM) of 30 and 75 is $150$.
Steps to Solve
- Prime Factorization of 30
First, we find the prime factorization of 30.
The factors of 30 are:
- 30 = 2 × 15
- 15 = 3 × 5
So, we can represent 30 as: $$ 30 = 2^1 × 3^1 × 5^1 $$
- Prime Factorization of 75
Next, we find the prime factorization of 75.
The factors of 75 are:
- 75 = 3 × 25
- 25 = 5 × 5
Thus, we can represent 75 as: $$ 75 = 3^1 × 5^2 $$
- Identify the Highest Powers of the Prime Factors
Now, we identify the highest powers of each prime factor from both factorizations:
- For the prime factor 2: $2^1$ (from 30)
- For the prime factor 3: $3^1$ (common in both)
- For the prime factor 5: $5^2$ (from 75)
- Calculate the Least Common Multiple (LCM)
Now we multiply the highest powers of all prime factors to find the LCM:
$$ LCM(30, 75) = 2^1 × 3^1 × 5^2 $$
Calculating it, we have:
- $2^1 = 2$
- $3^1 = 3$
- $5^2 = 25$
So:
$$ LCM(30, 75) = 2 × 3 × 25 $$
- Final Calculation
Now we compute the multiplication:
$$ 2 × 3 = 6 $$
Then,
$$ 6 × 25 = 150 $$
So the LCM of 30 and 75 is 150.
The least common multiple (LCM) of 30 and 75 is $150$.
More Information
The LCM is useful in finding a common denominator for fractions or when working with periodic events. In this problem, both numbers 30 and 75 can be used in various applications such as scheduling, and understanding their LCM helps manage timeframes.
Tips
- Forgetting to take the highest powers of the prime factors can lead to an incorrect LCM.
- Mixing up the prime factorization steps can cause confusion; it's important to clearly break it down.
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