Question

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

• A. 123555789
• B. 123457869
• C. 123456789
• D. 99999

Hint:

When a fixed digit-sum is required with distinct digits, first check the maximum achievable sum with $k$ digits and whether $0$ can appear; then sort the needed digits in increasing order for the smallest number.

Answer 1

The Correct Option is C

Step 1: Identify the target condition clearly
We are asked to find the smallest possible number whose digits add up to 45. Since the sum of digits is central, the maximum sum we can achieve with distinct digits 0–9 must be considered first. The largest possible digit sum with all 10 distinct digits is 0+1+2+...+9 = 45. But we want the minimum count of digits to achieve exactly 45.

Step 2: Test with 8 distinct digits
The largest 8-digit sum is 9+8+7+6+5+4+3+2 = 44, which is still less than 45. Hence, 8 digits are not enough. This shows that at least 9 distinct digits are required to reach 45.

Step 3: Check if 0 can be included in 9 distinct digits
Suppose 0 is included. Then the other 8 digits must be from 1–9. The maximum sum of those eight is 2+3+...+9 = 44. Adding 0 still gives 44, not 45. Therefore, including 0 makes it impossible to reach 45. So the required 9 digits must be exactly {1,2,3,4,5,6,7,8,9}.

Step 4: Form the smallest possible number
To minimize the actual number (not just the digit sum), we arrange the digits in increasing order. Thus the smallest number using digits 1 through 9 exactly once is: 123456789

Final Answer:
\[ \boxed{123456789} \]