The correct answer is (c).
The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of the original numbers. To find the LCM, you can use the prime factorization method.
To find the prime factorization of a number, you can write it as a product of its prime factors. For example, the prime factorization of 15 is $3\times5$.
To find the LCM of two or more numbers, you can find the prime factorization of each number and then multiply the prime factors together, taking the highest power of each prime factor. For example, the prime factorization of 15 is $3\times5$ and the prime factorization of 20 is $2^2\times5$. The LCM of 15 and 20 is therefore $2^2\times3\times5=60$.
In this case, the numbers are 15, 20, and $x$. The prime factorization of 15 is $3\times5$, the prime factorization of 20 is $2^2\times5$, and the prime factorization of $x$ is unknown. We know that the LCM of 15, 20, and $x$ is 180. Therefore, the prime factorization of $x$ must include at least the prime factors 2, 3, and 5. The only option that includes all three of these prime factors is (c), 18. Therefore, the smallest number $x$, such that the LCM of 15, 20, and $x$ is 180, is 18.
(a) 6 is not a multiple of 15 or 20. Therefore, it cannot be the LCM of 15, 20, and $x$.
(b) 9 is a multiple of 3, but it is not a multiple of 5. Therefore, it cannot be the LCM of 15, 20, and $x$.
(d) 45 is a multiple of 3 and 5, but it is not a multiple of 20. Therefore, it cannot be the LCM of 15, 20, and $x$.