1. We need to form 4-digit odd numbers using the digits $\{0,1,2,3,5,7\}$ with repetition allowed.

2. Conditions for a 4-digit odd number:

• The first digit cannot be $0$.
• The last digit must be odd.

3. Count choices position-wise.

First digit

Possible digits are $1,2,3,5,7$. So, number of choices = $5$

Last digit

To make the number odd, the last digit must be one of $1,3,5,7$ So, number of choices = $4$

Middle two digits

For each of the 2 middle places, any of the 6 digits can be used since repetition is allowed: $6 \text{ choices each}$

So total choices for the middle two digits: $6 \times 6 = 36$

4. Total number of such numbers: $5 \times 6 \times 6 \times 4 = 5 \times 36 \times 4 = 720$

5. Therefore, the total number of four-digit odd numbers is $\boxed{720}$

6. Comparing with the stored correct answer:

• Stored correct answer: D
• Derived answer: D

Hence, the derived answer agrees with the stored answer.