What is the smallest number with distinct digits whose digits add up to 45?

Determining the Smallest Number with Distinct Digits Summing to 45

This question requires us to identify the smallest integer that fulfills two conditions: all of its digits must be unique (distinct), and the sum of these digits must equal 45.

Understanding the Core Requirements

Let's clarify the conditions mentioned in the problem:

- Distinct Digits: The number must not contain any repeated digits. For instance, 123 is valid, but 131 is not, as '1' appears twice.
- Digit Sum of 45: Adding all the digits that make up the number must result in a total of 45. For example, for the number 258, the sum is $2+5+8=15$.
- Smallest Possible Number: Out of all the numbers that satisfy both the 'distinct digits' and 'sum of 45' criteria, we need to find the one with the minimum value. Generally, numbers with fewer digits are smaller than numbers with more digits. When comparing numbers with the same digit count, the number with smaller digits in the higher place values (leftmost) is considered smaller.

Identifying the Set of Digits

The first step is to determine which set of distinct digits sums up to 45. The digits we can use are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Let's calculate the maximum possible sum using distinct digits:

- The sum of all ten distinct digits (0 through 9) is: $$\sum _{i=0}^{9}i=0+1+2+3+4+5+6+7+8+9=45$$
- Since the required sum is precisely 45, using all ten digits (0 to 9) is a valid possibility.
- Could we achieve the sum of 45 using fewer than 10 distinct digits? Let's examine the maximum sum possible with 9 distinct digits. This occurs when we use the largest 9 digits available, which are 1 through 9: $$1+2+3+4+5+6+7+8+9=45$$ This combination also sums to 45.
- Now, consider using 8 distinct digits. The highest possible sum would be obtained by using the largest 8 digits: 2, 3, 4, 5, 6, 7, 8, 9. Their sum is $45-0-1=44$. Since 44 is less than 45, it is impossible to achieve a sum of 45 using only 8 distinct digits or fewer.

Based on this, the number must be formed using either the set of 9 digits {1, 2, 3, 4, 5, 6, 7, 8, 9} or the set of 10 digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

Constructing the Smallest Number from the Digit Sets

We now need to construct the smallest possible number for each valid set of digits.

Scenario 1: Using the 9 digits {1, 2, 3, 4, 5, 6, 7, 8, 9}

- To create the smallest number, these digits must be arranged in ascending order from left to right.
- Arranging them yields: 123456789.
- This results in a 9-digit number.

Scenario 2: Using the 10 digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

- To form the smallest number, we arrange the digits in ascending order. However, a standard number cannot begin with 0.
- Therefore, the smallest possible starting digit is 1.
- The remaining digits are {0, 2, 3, 4, 5, 6, 7, 8, 9}. To minimize the overall value of the number, these remaining digits should be placed in ascending order after the initial '1'.
- This arrangement results in the number 1023456789.
- This results in a 10-digit number.

Comparison of Scenarios: We compare the smallest number generated in Scenario 1 (123456789) with the smallest number from Scenario 2 (1023456789). A 9-digit number will always be smaller than a 10-digit number. Thus, 123456789 is the smaller of the two.

Conclusion: The smallest number possessing distinct digits that sum to 45 is 123456789.

Evaluating the Given Options

Let's review the choices provided in the question:Analysis of Options:

- Option 1 fails because the digits are not distinct (the digit 5 is repeated).
- Option 4 fails because the digits are not distinct (the digit 9 is repeated).
- Options 2 and 3 both satisfy the conditions: they use distinct digits, and the sum of their digits is 45.
- We need the smallest number. Comparing Option 2 (123457869) and Option 3 (123456789), we see that Option 3 is smaller. This is because when comparing them digit by digit from the left, they are identical up to the fifth digit. At the sixth position, Option 3 has a '6' while Option 2 has a '7'. Since 6 is less than 7, 123456789 is the smaller number.

Final Determination

The number 123456789 is the smallest number that uses distinct digits and whose digits sum to 45. This corresponds to Option 3.