If a number 67235x489 is divisible by 9, then find the value of x.

Options

A. 1
B. 3
C. 5
D. 7

Answer

To determine the value of \( x \) in the number \( 67235x489 \) such that it is divisible by 9, we can follow these steps:

### Step 1: Understand the Rule of Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Calculate the Sum of the Known Digits
We will first sum the known digits of the number \( 67235x489 \):
\[ 6 + 7 + 2 + 3 + 5 + x + 4 + 8 + 9 \]

Calculating the sum of the digits without \( x \):
\[ 6 + 7 = 13 \\ 13 + 2 = 15 \\ 15 + 3 = 18 \\ 18 + 5 = 23 \\ 23 + 4 = 27 \\ 27 + 8 = 35 \\ 35 + 9 = 44 \]
So, the sum of the digits is:
\[ 44 + x \]

### Step 3: Set Up the Equation for Divisibility
To find \( x \), we need \( 44 + x \) to be divisible by 9. We can express this as:
\[ 44 + x \equiv 0 \mod 9 \]

### Step 4: Calculate \( 44 \mod 9 \)
Now, we need to find \( 44 \mod 9 \):
\[ 44 \div 9 = 4 \quad \text{(which gives a quotient of 4 and a remainder of 8)} \]
Thus,
\[ 44 \equiv 8 \mod 9 \]

### Step 5: Solve for \( x \)
Now, we need to solve the equation:
\[ 8 + x \equiv 0 \mod 9 \]
This simplifies to:
\[ x \equiv -8 \mod 9 \]
Since \(-8\) is equivalent to \(1\) in modulo \(9\) (because \( -8 + 9 = 1 \)), we have:
\[ x \equiv 1 \mod 9 \]

### Step 6: Conclusion
The only single-digit value for \( x \) that satisfies this condition is:
\[ x = 1 \]

Thus, the value of \( x \) is \( 1 \).

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