Find the least common multiple (LCM) of 6 and 10.
A. 60
B. 20
C. 40
D. 30
The correct answer and explanation is:
The least common multiple (LCM) of 6 and 10 is 30.
Here’s how to find it:
- Prime Factorization:
First, break down both numbers into their prime factors:- 6 = 2 × 3
- 10 = 2 × 5
- Identify the Highest Powers of All Prime Factors:
Now, for each prime factor involved, select the highest power of that prime factor from both factorizations:- For 2: The highest power is 2¹ (from both 6 and 10).
- For 3: The highest power is 3¹ (from 6).
- For 5: The highest power is 5¹ (from 10).
- Multiply the Highest Powers Together:
Now, multiply these highest powers together:- LCM = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30.
Thus, the LCM of 6 and 10 is 30.
Explanation:
The LCM is the smallest number that is divisible by both of the given numbers. By using the prime factorization method, we ensure that all factors of both numbers are accounted for in the smallest possible multiple. The LCM is crucial in solving problems involving fractions, ratios, or any situation where you need a common multiple for addition or subtraction.
So, the correct answer is D. 30.