What is the value of `x` so that the seven digit number `91876x2` is divisible by `72`?
`X` का वह मान ज्ञात करें जिससे सात अंकों की संख्या `91876x2`, `72` से विभाजित हो जाए |

Options

A. 2
B. 7
C. 5
D. 3

Answer

To find the value of \( x \) such that the seven-digit number \( 91876x2 \) is divisible by \( 72 \), we need to ensure that it is divisible by both \( 8 \) and \( 9 \) since \( 72 = 8 \times 9 \).

### Step 1: Check divisibility by 8
A number is divisible by \( 8 \) if the number formed by its last three digits is divisible by \( 8 \). The last three digits of our number are \( 6x2 \).

**Form the number:**
The three-digit number is \( 6x2 \).

**Check values of \( x \):**
We will check for \( x = 0, 1, 2, \ldots, 9 \):

- For \( x = 0 \): \( 602 \div 8 = 75.25 \) (not divisible)
- For \( x = 1 \): \( 612 \div 8 = 76.5 \) (not divisible)
- For \( x = 2 \): \( 622 \div 8 = 77.75 \) (not divisible)
- For \( x = 3 \): \( 632 \div 8 = 79 \) (divisible)
- For \( x = 4 \): \( 642 \div 8 = 80.25 \) (not divisible)
- For \( x = 5 \): \( 652 \div 8 = 81.5 \) (not divisible)
- For \( x = 6 \): \( 662 \div 8 = 82.75 \) (not divisible)
- For \( x = 7 \): \( 672 \div 8 = 84 \) (divisible)
- For \( x = 8 \): \( 682 \div 8 = 85.25 \) (not divisible)
- For \( x = 9 \): \( 692 \div 8 = 86.5 \) (not divisible)

**Possible values of \( x \) for divisibility by \( 8 \):**
From our checks, \( x \) can be \( 3 \) or \( 7 \).

### Step 2: Check 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 \( 91876x2 \) is:

\[ 9 + 1 + 8 + 7 + 6 + x + 2 = 33 + x \]

**Check for \( x = 3 \):**
\[ 33 + 3 = 36 \quad (36 \div 9 = 4 \quad \text{divisible}) \]

**Check for \( x = 7 \):**
\[ 33 + 7 = 40 \quad (40 \div 9 \approx 4.44 \quad \text{not divisible}) \]

### Conclusion
The only value of \( x \) that satisfies both conditions (divisibility by \( 8 \) and \( 9 \)) is \( 3 \).

Thus, the value of \( x \) is \( \boxed{3} \).

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