Answer:
To form the least 5-digit number using the digits 1, 4, and 7 with the condition that the digit at the units place is 4 times the digit at the thousand place, we can proceed as follows:
Let's denote the digits as follows:
- Thousand place (T): \( x \)
- Units place (U): \( 4x \)
Given the digits are 1, 4, and 7, and \( 4x \) must be one of these digits.
1. **Selecting the digit for T and U**:
- Since \( 4x \) must be a single digit and one of 1, 4, or 7, we can try \( x = 1 \) because \( 4 \cdot 1 = 4 \) which is one of the given digits.
2. **Arranging the number**:
- Let \( x = 1 \), then \( T = 1 \) and \( U = 4 \).
3. **Choosing the remaining digits**:
- For the other three places (ten thousands, thousands, and hundreds), we can use the remaining digits 1, 4, and 7. To form the smallest number, arrange them in ascending order.
4. **Forming the number**:
- The smallest arrangement of the digits 1, 1, 4, 4, and 7 is \( 11447 \).
Therefore, the least 5-digit number using the digits 1, 4, and 7, with the condition that the digit at the units place is 4 times the digit at the thousand place, is \( \boxed{11447} \).
