Which digits should come in the place of * and #, respectively, if the 7-digit number 62684*# is divisible by both 8 and 5?

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Accepted Answer

Let's analyze the divisibility conditions:

1. For a number to be divisible by 5, its last digit must be 0 or 5. Here, the last digit is #. So, # can be 0 or 5.

Let's check both cases:

Case 1: # = 0
So, the number is 62684*0.

2. For a number to be divisible by 8, the last three digits must form a number divisible by 8. The last three digits are 4*0.
Let’s try all possible values for * (from 0 to 9):

- 440 ÷ 8 = 55 (divisible)
- 450 ÷ 8 = 56.25 (not divisible)
- 460 ÷ 8 = 57.5 (not divisible)
- 470 ÷ 8 = 58.75 (not divisible)
- 480 ÷ 8 = 60 (divisible)
- 490 ÷ 8 = 61.25 (not divisible)

So, possible values for * are 4 and 8 (since 440 and 480 are divisible by 8).

But only 440 is among the options (since * must be a single digit and 4*0 means * = 4).

Now, check the options:
- Option 2: 4 and 0 (i.e., * = 4, # = 0) gives last three digits 440, which is divisible by 8 and ends in 0 (divisible by 5).

Case 2: # = 5
Last digit is 5, so last three digits are 4*5. Try all possible *:
- 405 ÷ 8 = 50.625 (not divisible)
- 415 ÷ 8 = 51.875 (not divisible)
- 425 ÷ 8 = 53.125 (not divisible)
- 435 ÷ 8 = 54.375 (not divisible)
- 445 ÷ 8 = 55.625 (not divisible)
- 455 ÷ 8 = 56.875 (not divisible)
- 465 ÷ 8 = 58.125 (not divisible)
- 475 ÷ 8 = 59.375 (not divisible)
- 485 ÷ 8 = 60.625 (not divisible)
- 495 ÷ 8 = 61.875 (not divisible)

None of these are divisible by 8.

Therefore, the only possible answer is * = 4 and # = 0.

Final answer: Option 2 (4 and 0).

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