Find the least common multiple (LCM ) for the following pair of numbers. 7 and 8 The least common multiple of 7 and 8 is
The Correct Answer and Explanation is:
The least common multiple (LCM) of 7 and 8 is 56.
Explanation:
To find the LCM of two numbers, we follow these steps:
- Prime Factorization:
Start by finding the prime factorization of each number:- 7 is a prime number, so its prime factorization is simply 7.
- 8 is a power of 2, so its prime factorization is 2 × 2 × 2 or 2³.
- Identify the highest powers of all prime factors:
For the LCM, we need to take the highest powers of all the prime factors that appear in the prime factorizations of both numbers.- For 7, the prime factor is 7 (highest power is 7¹).
- For 8, the prime factor is 2 (highest power is 2³).
- Multiply the highest powers of each prime factor:
Now, multiply the highest powers of each prime factor: LCM=71×23=7×8=56LCM = 7^1 × 2^3 = 7 × 8 = 56LCM=71×23=7×8=56
Thus, the LCM of 7 and 8 is 56.
Why this works:
The LCM is the smallest number that is divisible by both 7 and 8. By using prime factorization, we ensure that the LCM includes each factor at its highest power. This ensures that the result will be divisible by both original numbers. In this case, 56 is the smallest number that both 7 and 8 can divide without leaving a remainder, which confirms it as the LCM.
