What is the Smallest 4-Digit Number with No Repeated Digits?

When asked to identify the smallest 4-digit number with no repeated digits, the commonly accepted answer is 1023. This choice is based on the usage of the smallest non-zero digit followed by the next smallest unique digits in ascending order.

The Answer: 1023

The smallest 4-digit number with no repeated digits is 1023. This answer adheres to the convention that the number must not only be a 4-digit number but also have each digit appear exactly once. Starting with the smallest non-zero digit and proceeding to the smallest available digits ensures the formation of the number 1023.

Alternative Considerations

Some might argue that 0123 could also be considered a valid answer based on different interpretations of the rules. However, most conventional standards do not permit 0123 as the first digit is not considered a 4-digit number. There is no explicit rule stating that the first digit must be non-zero, but such interpretation is generally accepted in mathematical and digital contexts.

Forming the Smallest 4-Digit Number

Four unique digits for a 4-digit number are 0, 1, 2, and 3. To form the smallest 4-digit number, we place the most significant digit as 1 (since 0 would not constitute a 4-digit number), followed by the remaining digits in ascending order: 0, 2, and 3. Thus, the smallest 4-digit number with no digits repeated is 1023.

Further Insights

When considering different bases or interpretations, several variations can arise. For instance, if we allow leading zeros, 0123 would be valid but generally not considered in standard interpretations. Should we define the number strictly as a base-10 number, 1023 remains the smallest valid option.

For a different level of complexity, if we require no two consecutive digits to be the same, then the answer would be 1010 in base 2 (or 10 in base 10). If all 4 digits must be different, the smallest number in base 4 is 1320 (or 120 in base 10).

Conclusion

The smallest 4-digit number with no repeated digits, following the standard interpretation, is 1023. This answer is straightforward and aligns with common mathematical and digital conventions. Understanding these nuances can help in addressing similar questions and ensuring clarity in communication and problem-solving.