$\begin{array}{r} 8532 \\ - 2358 \\ \hline 6174 \end{array}$ Takes 7 rounds Q3 We are .......... $50{,}000$ ? 5 digits number between $35{,}000$ and $75{,}000$ . All digits are odd: $1, 3, 5, 7, 9$ . Largest number = $73999$ Smallest number = $35111$ Number closest to $50{,}000 = 49{,}999$

Answer: Largest number = $73{}999$ , Smallest number = $35{}111$ , Number closest to $50{}000 = 51{}111$

Step-by-step solution

Step 1: Find the largest number

To make the number strictly less than $75{}000$ using only odd digits $\{1, 3, 5, 7, 9\}$ , the ten-thousands digit can be at most $7$ . Since the number must be less than $75{}000$ , if the first digit is $7$ , the thousands digit cannot be $5$ or higher, so the maximum possible odd digit for the thousands place is $3$ . To maximize the rest of the number, the remaining digits in the hundreds, tens, and units places are all chosen as $9$ .

Step 2: Find the smallest number

To make the number strictly greater than $35{}000$ using only odd digits $\{1, 3, 5, 7, 9\}$ , the smallest possible ten-thousands digit is $3$ . If the first digit is $3$ , the thousands digit must be at least $5$ (since choosing $1$ or $3$ would make the number smaller than $35{}000$ ). Thus, the thousands digit is $5$ . To minimize the remaining places, we choose the smallest odd digit, which is $1$ , for the hundreds, tens, and units places.

Step 3: Find the number closest to 50,000

We need all digits to be odd. A number just below $50{}000$ would have to start with $3$ (since $4$ is even), making the closest number below $50{}000$ equal to $39{}999$ , which is at a distance of $10{}001$ . A number just above $50{}000$ can start with $5$ , followed by the smallest odd digits, giving $51{}111$ , which is at a distance of only $1{}111$ . Comparing the two distances, $1{}111 < 10{}001$ , so the number closest to $50{}000$ is $51{}111$ . Note that $49{}999$ is invalid because its first digit $4$ is even.

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