What is the least common multiple (LCM) of 8 and 14?
The correct answer and explanation is:
The least common multiple (LCM) of 8 and 14 is 56.
Explanation:
To find the LCM of two numbers, follow these steps:
- Prime Factorization:
Start by finding the prime factors of each number.- For 8:
8 is a power of 2, so its prime factorization is: 8=238 = 2^3 - For 14:
14 can be broken down into prime factors: 14=2×714 = 2 \times 7
- For 8:
- Identify the Highest Powers of Each Prime Factor:
The LCM is found by multiplying the highest powers of all the primes that appear in the prime factorizations.- For 2, the highest power is 232^3 (from 8).
- For 7, the highest power is 717^1 (from 14).
- Multiply the Highest Powers:
Multiply these highest powers together: LCM=23×71=8×7=56LCM = 2^3 \times 7^1 = 8 \times 7 = 56 - Conclusion:
The least common multiple of 8 and 14 is 56. This is the smallest number that is divisible by both 8 and 14. The LCM can be used in various contexts like solving problems involving fractions, ratios, and periodic events, where you want to find the smallest common time or value.
By finding the LCM, you can ensure that events or actions that happen at intervals of 8 and 14 units of time, for example, will coincide every 56 units of time. This is the power of LCM in practical applications like scheduling, event planning, or dealing with problems in number theory.