List the first six common multiples of 6 and 9

List the first six common multiples of 6 and 9.

The Correct Answer and Explanation is :

To find the first six common multiples of 6 and 9, we need to identify the least common multiple (LCM) of these two numbers first.

Finding the LCM of 6 and 9:

1. Prime Factorization:

• The prime factorization of 6 is (2^1 \times 3^1).
• The prime factorization of 9 is (3^2).

1. Identifying the LCM:

• The LCM is found by taking the highest power of each prime number present in the factorizations.
• For the prime number 2, the highest power present is (2^1) (from 6).
• For the prime number 3, the highest power is (3^2) (from 9).
• Thus, the LCM is (2^1 \times 3^2 = 2 \times 9 = 18).

First Six Common Multiples:

Now that we have the LCM, we can find the common multiples by multiplying the LCM by the integers from 1 to 6:

1. (1 \times 18 = 18)
2. (2 \times 18 = 36)
3. (3 \times 18 = 54)
4. (4 \times 18 = 72)
5. (5 \times 18 = 90)
6. (6 \times 18 = 108)

Result:

The first six common multiples of 6 and 9 are:

• 18, 36, 54, 72, 90, and 108.

Explanation:

Common multiples are the numbers that both given numbers can divide without leaving a remainder. By finding the LCM, we can generate these multiples systematically. Since 18 is the smallest number that both 6 and 9 divide into evenly, it serves as the foundational common multiple. Multiplying the LCM by successive integers gives us the subsequent common multiples. This process is crucial in various mathematical applications, including solving problems related to fractions, ratios, and algebra, ensuring that operations involving multiple numbers are simplified effectively. By understanding how to derive common multiples, one can tackle a range of mathematical challenges with greater ease.