Final answer:
To find the LCM of two numbers, both numbers must be factors of the LCM. Since the LCM is 108 and one of the numbers is 54, the second number must also be a factor of 108 but not greater than 54. The maximum factor of 108 that meets these criteria is 27.
Step-by-step explanation:
The question asks us to find the maximum value of the second number given that the least common multiple (LCM) of two numbers is 108 and the greater number is 54. To find the LCM of two numbers, you need to multiply the highest power of all the prime factors present in both numbers. Since 54 is one of the numbers and also a factor of 108, the second number must be a factor of 108 as well so as to have 108 as the LCM.
Out of the given options, the one that is a factor of 108 and not greater than 54 is 27 (Option D). Therefore, the maximum value of the second number that, along with 54, gives an LCM of 108 is 27.