Find the number of 5 - digit numbers divisible by 5 that can be formed using the digits 0, 1, 2, 3, 4, 5 when repetition is allowed.

To determine the number of 5-digit numbers divisible by 5 that can be formed using the digits 0, 1, 2, 3, 4, and 5 with repetition allowed, we need to focus on the last digit and the first digit of the number.

Identifying the Last Digit

A number is divisible by 5 if its last digit is either 0 or 5. Therefore, we have two options for the last digit:

• Last digit = 0
• Last digit = 5

Calculating the First Digit

The first digit of a 5-digit number cannot be 0 (as it would then be a 4-digit number). Thus, the possible choices for the first digit are 1, 2, 3, 4, or 5, giving us 5 options.

Choosing the Middle Digits

The three middle digits can be any of the six available digits (0, 1, 2, 3, 4, 5), since repetition is allowed. Therefore, for each of the three middle positions, we have 6 choices.

Putting It All Together

Now, we can calculate the total number of valid 5-digit numbers:

• If the last digit is 0:
  • First digit: 5 choices
  • Middle digits: 6 choices each for 3 positions

  Total for last digit 0 = 5 × 6 × 6 × 6 = 5 × 216 = 1080

• If the last digit is 5:
  • First digit: 5 choices
  • Middle digits: 6 choices each for 3 positions

  Total for last digit 5 = 5 × 6 × 6 × 6 = 5 × 216 = 1080

Final Calculation

Adding both cases together gives:

Total = 1080 (last digit 0) + 1080 (last digit 5) = 2160

Thus, the total number of 5-digit numbers divisible by 5 that can be formed is 2160.