The least common multiple (LCM) is the smallest positive integer that is divisible by all the given numbers. In this case, we are given that the LCM of 18, 45, and x is 180. We need to determine which of the given options can be the possible value of x.

To find the LCM of the numbers, we can start by finding the prime factorization of each number:

- 18: 2 x 3²
- 45: 3² x 5
- x: Prime factorization unknown

Next, we need to determine the highest power of each prime factor that appears in the prime factorization of the LCM, which is 180:

- 180: 2² x 3² x 5

Comparing the prime factorizations, we can see that the LCM includes the prime factors 2, 3, and 5, and the highest powers of each prime factor are 2², 3², and 5¹.

- The prime factorization of x is unknown, but it must include at least the prime factors 2, 3, and 5, with powers less than or equal to 2², 3², and 5¹, respectively.

Now, let's analyze each option:

a) 27: The prime factorization of 27 is 3³. Since the LCM can only have a maximum power of 3² for the prime factor 3, 27 cannot be the value of x.

b) 36: The prime factorization of 36 is 2² x 3². This satisfies the conditions for the prime factors 2 and 3, with powers less than or equal to 2² and 3², respectively. Therefore, 36 can be the value of x.

c) 54: The prime factorization of 54 is 2 x 3³. Since the LCM can only have a maximum power of 3² for the prime factor 3, 54 cannot be the value of x.

d) 92: The prime factorization of 92 is 2² x 23. While 92 includes the prime factor 2, it does not include the prime factors 3 and 5 that are necessary for the LCM. Therefore, 92 cannot be the value of x.

Therefore, the only possible value for x is b) 36.