If the 7-digit number 134x58y is divisible by 72, then the value of (2x + y) is

Question

Accepted Answer

To determine the value of $2x + y$ for the 7-digit number 134x58y to be divisible by 72, we need to ensure that the number is divisible by both 8 and 9, as 72 is the product of these two numbers.

**Step 1: Divisibility by 8**

A number is divisible by 8 if its last three digits form a number that is divisible by 8. In this case, the last three digits are "58y". We need to find a value for $y$ such that 58y is divisible by 8.

Let's test the possible values for $y$ (0 through 9):

- 580 ÷ 8 = 72.5 (not divisible)
- 581 ÷ 8 = 72.625 (not divisible)
- 582 ÷ 8 = 72.75 (not divisible)
- 583 ÷ 8 = 72.875 (not divisible)
- 584 ÷ 8 = 73 (divisible)
- 585 ÷ 8 = 73.125 (not divisible)
- 586 ÷ 8 = 73.25 (not divisible)
- 587 ÷ 8 = 73.375 (not divisible)
- 588 ÷ 8 = 73.5 (not divisible)
- 589 ÷ 8 = 73.625 (not divisible)

From this, we see that $y = 4$ makes 584 divisible by 8.

**Step 2: Divisibility by 9**

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of 134x584 is $1 + 3 + 4 + x + 5 + 8 + 4 = 25 + x$.

We need $25 + x$ to be divisible by 9. Testing values for $x$:

- If $x = 2$, then $25 + 2 = 27$, which is divisible by 9.

Thus, $x = 2$ and $y = 4$ satisfy both conditions for divisibility by 8 and 9.

**Calculating $2x + y$:**

Given $x = 2$ and $y = 4$:

$$2x + y = 2(2) + 4 = 4 + 4 = 8$$

Therefore, the value of $2x + y$ is 8, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (6):** This would imply a different combination of $x$ and $y$ that does not satisfy the divisibility rules for both 8 and 9.
- **Option 2 (7):** Similarly, this value does not result from any valid combination of $x$ and $y$ that meets the divisibility criteria.
- **Option 4 (9):** This would also require a different set of values for $x$ and $y$ that do not satisfy the conditions for divisibility by both 8 and 9.

Thus, Option 3 is the only correct answer, as it satisfies all the conditions for divisibility by 72.