What is the smallest 3-digit number with unique digits?

The smallest 3-digit number with unique digits is 102. To understand why this is the case, let's break down the criteria for a 3-digit number and what it means for the digits to be unique.

Understanding 3-Digit Numbers

A 3-digit number ranges from 100 to 999. The first digit cannot be zero because that would make it a 2-digit number. Therefore, the smallest possible first digit is 1.

Choosing the Digits

Since we want the smallest number, we start with 1 as the first digit. Now, we need to choose the next two digits, ensuring they are unique and different from the first digit.

• The first digit is 1.
• The second digit can be the smallest available digit, which is 0.
• For the third digit, we need to select the next smallest digit that hasn't been used yet. The smallest available digit after 1 and 0 is 2.

Forming the Number

Putting these digits together, we get the number 102. This number meets all the criteria:

• It is a 3-digit number.
• All digits (1, 0, and 2) are unique.

Verifying Other Options

To ensure that 102 is indeed the smallest, let’s consider other combinations starting with 1:

• 101 is not valid because the digits are not unique.
• 103 uses the digits 1, 0, and 3, which is larger than 102.
• 104, 105, and so on, also yield larger numbers.

Conclusion

After evaluating the possibilities, it’s clear that 102 is the smallest 3-digit number with unique digits. This method of systematically selecting the smallest digits ensures that we arrive at the correct answer efficiently.