Four numbers are proportional to 10 : 12 : 15 : 18. Given that their highest common factor (HCF) is 3, determine the least common multiple (LCM) of the actual four numbers.

Introduction / Context:
This problem tests the relationship between a set of numbers given in a ratio and their actual values when a highest common factor (HCF) is specified. Once the actual numbers are reconstructed from the ratio using the given HCF, the least common multiple (LCM) can be computed by prime factorization or by reasoning about maximum prime powers.

Given Data / Assumptions:

• Ratio: 10 : 12 : 15 : 18.
• HCF of the four actual numbers is 3.
• All quantities are positive integers.

Concept / Approach:
If numbers are in the ratio k1 : k2 : k3 : k4 and the HCF of the actual numbers is h, then the actual numbers can be taken as h * k1, h * k2, h * k3, h * k4 provided that HCF(k1, k2, k3, k4) = 1. Then compute the LCM of these four actual integers using the highest power of each prime that appears across them.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 420, 620, 680 miss at least one required prime power (e.g., 3^3) and will not be divisible by all four numbers simultaneously.

Common Pitfalls:

• Using the LCM of the ratio parts and forgetting to multiply by the HCF. Here, because the HCF is incorporated in the actual numbers already, the prime-power method is safest.

Final Answer: