Answer :
To solve this problem, we need to find how many 4-digit numbers consist entirely of odd digits and are divisible by 5.
Understanding Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5. However, since we are asked to use only odd digits, the last digit must be 5.
Identifying Possible Digits: The odd digits are: 1, 3, 5, 7, and 9.
Setting Conditions for the Number: Since it’s a 4-digit number, the first digit cannot be zero. Given these criteria, each of the digits can only be one of the odd digits. The structure of the number can be represented as 'ABCD', where each letter stands for a digit and 'D' must be 5.
Calculating the Possibilities:
- For the first digit 'A', you have 5 options: 1, 3, 5, 7, 9.
- For the second digit 'B', you also have 5 options: 1, 3, 5, 7, 9.
- For the third digit 'C', you again have 5 options: 1, 3, 5, 7, 9.
- For the last digit 'D', you have only 1 option: 5 (since the number must be divisible by 5).
Calculating Total Combinations: Multiply the number of choices for each digit to find the total number of combinations.
[tex]\text{Total number of 4-digit numbers} = 5 \times 5 \times 5 \times 1 = 125[/tex]
Hence, there are 125 such 4-digit numbers.
The correct answer is option (a) 125.