If an 8-digit number 30x0867y, 88 से विभाज्य है, तो (3x + y) का मान क्या है ?

Question

Accepted Answer

To determine why Option 2 is the correct answer for the value of $3x + y$ when the 8-digit number $30x0867y$ is divisible by 88, we need to understand the divisibility rules for 88. A number is divisible by 88 if it is divisible by both 8 and 11.

**Step 1: Divisibility by 8**

For a number to be divisible by 8, its last three digits must be divisible by 8. In this case, the last three digits are $67y$.

- If $y = 0$, $670$ is not divisible by 8.
- If $y = 1$, $671$ is not divisible by 8.
- If $y = 2$, $672$ is divisible by 8.
- If $y = 3$, $673$ is not divisible by 8.
- If $y = 4$, $674$ is not divisible by 8.
- If $y = 5$, $675$ is not divisible by 8.
- If $y = 6$, $676$ is not divisible by 8.
- If $y = 7$, $677$ is not divisible by 8.
- If $y = 8$, $678$ is not divisible by 8.
- If $y = 9$, $679$ is not divisible by 8.

Thus, $y = 2$ is the only value that satisfies the divisibility rule for 8.

**Step 2: Divisibility by 11**

For a number to be divisible by 11, the difference between the sum of its digits in odd positions and the sum of its digits in even positions must be a multiple of 11.

The number is $30x08672$.

- Sum of digits in odd positions: $3 + x + 8 + 7 = 18 + x$
- Sum of digits in even positions: $0 + 0 + 6 + 2 = 8$

The difference is $(18 + x) - 8 = 10 + x$.

For divisibility by 11, $10 + x$ must be a multiple of 11. The possible value for $x$ that satisfies this condition is $x = 1$, because $10 + 1 = 11$, which is a multiple of 11.

**Conclusion**

With $x = 1$ and $y = 2$, the expression $3x + y$ becomes:

$$3(1) + 2 = 3 + 2 = 5$$

Therefore, the correct answer is Option 2: 5.

**Explanation of Incorrect Options:**

1. **Option 1 (4):** This would imply a different combination of $x$ and $y$ that does not satisfy the divisibility rules for both 8 and 11.
2. **Option 3 (6):** This would also imply a different combination of $x$ and $y$ that does not satisfy the divisibility rules for both 8 and 11.
3. **Option 4 (7):** This would imply a different combination of $x$ and $y$ that does not satisfy the divisibility rules for both 8 and 11.

Thus, Option 2 is the only correct answer.