Final answer:
To find the least possible value of a number with a GCD of 30 and an LCM of 900, given the other two numbers are 60 and 150, we must select a number that is a multiple of 30 and also a factor of 900. The smallest such number that meets these criteria is 180.
Step-by-step explanation:
The question provided requires an understanding of the relationship between the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of numbers. We are also given two of the numbers, which are 60 and 150, and we need to find the least possible value of the third number that combines with 60 and 150 to have a GCD of 30 and an LCM of 900.
The GCD of any two numbers divides those numbers and any potential third number. Given that the GCD is 30, this implies that the third number is a multiple of 30. If we list the multiples of 30 starting with the least multiples we have 30, 60, 90, etc. But since the third number needs to have a GCD of 30 with 60 and 150, which both are already multiples of 30, this number also needs to be a multiple of 30 and not introduce any new prime factors that could alter the GCD.
The LCM of numbers is the smallest number that all the numbers can divide into without leaving a remainder. Given that the LCM of the three numbers is 900, and we know that 60 and 150 are two of those numbers, the third number must be a factor of 900 and must combine with 60 and 150 to reach that LCM when considering their prime factorizations. The smallest such number that does this while also being a multiple of 30 is 180. Since both the GCD requirement and the LCM requirement are satisfied by 180, the answer is (A) 180.