Explain how to find the LCM of 8 and 10
To find the Least Common Multiple (LCM) of two numbers, like 8 and 10, we can use several methods. I’ll walk you through one method that’s fairly straightforward—prime factorization—and then we’ll check our result.
Step 1: Prime Factorization
We start by breaking down each number into its prime factors.
- For 8, we can express it as:
8=2×2×28 = 2 \times 2 \times 28=2×2×2
This means the prime factorization of 8 is 232^323. - For 10, we can express it as:
10=2×510 = 2 \times 510=2×5
The prime factorization of 10 is 21×512^1 \times 5^121×51.
Step 2: Identify the Highest Powers of All Prime Factors
To find the LCM, we need to take the highest power of each prime factor that appears in either number.
- The prime factor 2 appears in both 8 and 10, but the highest power is 232^323 (from 8).
- The prime factor 5 only appears in 10, so we take 515^151.
Step 3: Multiply the Highest Powers
Now, multiply the highest powers of all the primes together to find the LCM:LCM=23×51=8×5=40LCM = 2^3 \times 5^1 = 8 \times 5 = 40LCM=23×51=8×5=40
Conclusion
So, the LCM of 8 and 10 is 40. This means that 40 is the smallest number that both 8 and 10 divide evenly into.
This method of prime factorization ensures that we account for all the factors while ensuring we only take the necessary multiples. You can verify this by listing the multiples of each number:
- The multiples of 8 are: 8, 16, 24, 32, 40, …
- The multiples of 10 are: 10, 20, 30, 40, …
As you can see, 40 is the first common multiple, which confirms our answer.
