Smallest 6-Digit Number Formation

We need to find the smallest 6-digit number using only the digits 0, 4, and 6, ensuring each digit appears at least once.

• A 6-digit number cannot start with 0. The smallest possible non-zero starting digit is 4.
• To make the number as small as possible, the remaining digits should be arranged in ascending order.
• The number must contain 0 and 6. We have used 4 initially. We need 5 more digits.
• To minimize the value, we use the smallest available digits (0, 4, 6) after the first digit. We prioritize 0s.
• We must include at least one 0 and one 6. The remaining required digits (3) can be filled with 0s.
• Therefore, the sequence is 4 followed by 0, 0, 0, 0, 6.
• The smallest number is 400006.

Largest 6-Digit Number Formation

We need to find the largest 6-digit number using only the digits 0, 4, and 6, ensuring each digit appears at least once.

• To maximize the number, the first digit should be the largest available digit, which is 6.
• To make the number as large as possible, the remaining digits should be arranged in descending order.
• The number must contain 0 and 4. We have used 6 initially. We need 5 more digits.
• To maximize the value, we use the largest available digits (0, 4, 6) after the first digit. We prioritize 6s.
• We must include at least one 0 and one 4. The remaining required digits (3) can be filled with 6s.
• Therefore, the sequence is 6 followed by 6, 6, 6, 4, 0.
• The largest number is 666640.

Sum Calculation

To find the final answer, we sum the smallest and the largest numbers found.

• Smallest Number = 400006
• Largest Number = 666640
• Sum = Smallest Number + Largest Number
• $ \text{Sum} = 400006 + 666640 $
• $ \text{Sum} = 1066646 $

The sum of the smallest and the largest 6-digit numbers formed using digits 0, 4, 6 (each appearing at least once) is 1066646.