Final answer:
The smallest number of months that must pass before Planets A, B, and C line up again is 126 months, which is the least common multiple of their orbital periods.
Step-by-step explanation:
The question is asking for the next time that planets A, B, and C will align in a straight line. To solve this, we need to find the least common multiple (LCM) of their orbital periods, which are 3, 7, and 18 months respectively. The LCM of these numbers gives us the smallest number of months that must pass before all three planets line up again.
The LCM of 3 and 7 is 21 since 3 and 7 are relatively prime (they have no common factors other than 1). The LCM of 21 and 18 is found by breaking down the numbers into their prime factors: 21 is 3 times 7, and 18 is 2 times 3 squared. The highest power of all prime factors present in the numbers must be used to calculate the LCM, which in this case is 2 times 3 squared times 7 (because 3 appears as 3 squared in the factorization of 18, and as 3 in the factorization of 21). Thus, the LCM is 2 times 9 times 7, equating to 126. Therefore, it will take 126 months for planets A, B, and C to align again in the same straight line.