(b) 6, 8, 12
Multiples of 6 :
Multiples of 8 :
Multiples of 12:
Common multiples of 6, 8 and 12 are
LCM of 6, 8 and 12
Answers
Answer:
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Answer:
Step 1: Determine the prime factors of each number.
Positive whole numbers fall into two categories: prime or composite. Prime numbers have only two positive factors: 1 and the number itself. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 27, etc. Note that 2 is the only even prime number.
Composite numbers have positive factors other than 1 and the number itself. For example, 4 is a composite number because its positive factors include 1, 2, and 4. Six is composite because its factors include 1, 2, 3, and 6.
Every composite number can be written as a unique product of primes. This unique product is known as the prime factorization of the number.
You have provided three composite numbers. Here is each number’s prime factorization:
6 = 2 x 3
8 = 2 x 2 x 2
12 = 2 x 2 x 3
Step 2: Identify the primes that must be used in the construction of the LCM.
As you can see, there are only two primes used in these numbers’ prime factorizations: 2 and 3.
Step 3: Determine the multiplicity of each prime in the LCM’s prime factorization.
Look at the prime factorizations of the original three numbers. 2 appears once in the prime factorization of 6; thrice in the prime factorization of 8, and twice in the prime factorization of 12. The LCM’s prime factorization must include three 2s, since that is the minimum number of factors of 2 necessary to cover all of the original numbers.
Also, 3 appears once in the prime factorization of 6, once in the prime factorization of 12, and not at all in the prime factorization of 8. So, the LCM’s prime factorization must have one 3.
The prime factorization of the LCM is therefore:
2 x 2 x 2 x 3
Step 4: Determine the LCM by multiplying the primes.
2 x 2 x 2 x 3 = 24. The LCM of 6, 8, and 12 is 24.
Step 5: Using the largest of the original numbers, check your work by confirming that there are no other common multiples below the LCM you calculated.
The largest of the original numbers is 12. There are no other multiples of 12 between 12 and 24, and 12 is not divisible by 8, so 24 is the LCM. Check!