Problem Analysis: Smallest Number with Distinct Digits Summing to 45
The objective is to find the smallest integer that satisfies two conditions:
- Its digits must all be different (distinct).
- The sum of its digits must equal 45.
Identifying the Digits
The set of available digits is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Let's calculate the sum of all these ten distinct digits:
Sum = $0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$.
This sum matches the target sum required by the question.
Constructing the Smallest Number
To find the smallest number, we prioritize using the fewest possible digits. If multiple combinations have the same number of digits, we arrange them to form the smallest value (smallest digits at the beginning).
Smallest Number Using 10 Digits
Using all ten distinct digits (0-9) results in a sum of 45. To form the smallest possible 10-digit number, we arrange these digits in ascending order, ensuring the first digit is not zero. The smallest arrangement is 1023456789.
Smallest Number Using 9 Digits
Consider using 9 distinct digits. The sum of digits 1 through 9 is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$. This set of digits {1, 2, 3, 4, 5, 6, 7, 8, 9} sums to 45.
The smallest number formed using these 9 digits is achieved by arranging them in ascending order: 123456789.
Could any other set of 9 distinct digits sum to 45? If we remove any digit from {0, 1, ..., 9} except 0, the sum is less than 45. For example, removing 1 gives a sum of 44. Removing 9 gives a sum of 36. The only way to get 9 distinct digits summing to 45 is by using the digits 1 through 9.
Could we use 8 or fewer digits? The maximum sum possible with 8 distinct digits is $9+8+7+6+5+4+3+2 = 44$, which is less than 45. Therefore, the minimum number of digits required is 9 (using 1-9) or 10 (using 0-9).
Comparison and Final Answer
We compare the smallest numbers formed in each case:
- Smallest 10-digit number: 1023456789
- Smallest 9-digit number: 123456789
A 9-digit number is always smaller than a 10-digit number. Therefore, 123456789 is the smallest number with distinct digits whose digits sum to 45.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
