Answer:
Let's break down the problem into steps:
Step 1: Find the common factors of 45 and the given digits.
The prime factorization of 45 is 3 * 3 * 5. Therefore, the common factors of 45 and the given digits are 3 and 5.
Step 2: Determine the first digit of the two 3-digit numbers.
Since the common factors are 3 and 5, the first digit of both numbers must be either 3 or 5. However, the LCM must also be under 3000, so it's best to start with 3.
Step 3: Determine the second digit of the two 3-digit numbers.
The second digit can be any of the given digits (0, 1, 6, 7, or 8) except for 5 because it is already a common factor.
Step 4: Determine the third digit of the two 3-digit numbers.
The third digit can be any of the given digits (0, 1, 6, 7, or 8) except for 5 because it is already a common factor.
Step 5: Calculate the LCM of the two numbers.
Multiply the two numbers together and divide by the HCF to find the LCM. If the LCM is under 3000, we have found our solution.
For example, the two 3-digit numbers could be:
345 and 108
HCF = 45
LCM = (345 * 108) / 45 = 828
Step 6: Repeat steps 2-5 with different combinations of digits until you find a pair of 3-digit numbers with an HCF of 45 and LCM under 3000.
Note: It is possible that there are no combinations of digits that satisfy both criteria. In that case, it would be helpful to double-check the given digits and the conditions of the problem.
Explanation: