Understanding Divisibility Rules

To solve this problem, we need to apply the divisibility rules for 8 and 5.

• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Applying Rules to 2893#$

The number is $2893\#\$. We need it to be divisible by both 8 and 5.

1. Divisibility by 5: The last digit, represented by '$', must be 0 or 5.
2. Divisibility by 8: The number formed by the last three digits, $3\#\$, must be divisible by 8.

Testing Possible Values

Let's consider the two possibilities for '$' based on the divisibility by 5 rule:

Case 1: The last digit ($) is 5.

If '$' = 5, the number is $2893\#5$. For this number to be divisible by 8, the last three digits, $3\#5$, must be divisible by 8. However, any number ending in 5 is odd. Since all multiples of 8 are even, $3\#5$ cannot be divisible by 8. Therefore, '$' cannot be 5.

Case 2: The last digit ($) is 0.

If '$' = 0, the number is $2893\#0$. For this number to be divisible by 8, the last three digits, $3\#0$, must be divisible by 8.

We need to find a digit for '#' such that $3\#0$ is divisible by 8. Let's test values:

• If # = 0, $300 \div 8 = 37.5$ (Not divisible)
• If # = 1, $310 \div 8 = 38.75$ (Not divisible)
• If # = 2, $320 \div 8 = 40$ (Divisible by 8)
• If # = 3, $330 \div 8 = 41.25$ (Not divisible)
• If # = 4, $340 \div 8 = 42.5$ (Not divisible)
• If # = 5, $350 \div 8 = 43.75$ (Not divisible)
• If # = 6, $360 \div 8 = 45$ (Divisible by 8)
• If # = 7, $370 \div 8 = 46.25$ (Not divisible)
• If # = 8, $380 \div 8 = 47.5$ (Not divisible)
• If # = 9, $390 \div 8 = 48.75$ (Not divisible)

The possible values for '#' are 2 and 6 when '$' is 0. This gives us two possible pairs for (#, $): (2, 0) and (6, 0).

Identifying the Correct Option

We are looking for the pair (#, $) that satisfies the conditions. Let's check the given options:

• Option 1: (2, 0). This fits our findings. The number is 289320. It ends in 0 (divisible by 5) and 320 is divisible by 8.
• Option 2: (0, 0). The number is 289300. It ends in 0 (divisible by 5) but 300 is not divisible by 8.
• Option 3: (2, 2). The number is 289322. It does not end in 0 or 5 (not divisible by 5).
• Option 4: (0, 2). The number is 289302. It does not end in 0 or 5 (not divisible by 5).

Therefore, the digits that would come in the place of # and $, respectively, are 2 and 0.