Find the value of x for which the number x806 is divisible by 9. Also find the number.

Answer

To find the value of \( x \) for which the number \( x806 \) is divisible by 9, we can follow these steps:

### Step 1: Understand the divisibility rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Write down the digits of the number
The digits of the number \( x806 \) are \( x, 8, 0, 6 \).

### Step 3: Calculate the sum of the digits
The sum of the digits can be expressed as:
\[ x + 8 + 0 + 6 = x + 14 \]

### Step 4: Set up the equation for divisibility
For \( x806 \) to be divisible by 9, the sum \( x + 14 \) must be a multiple of 9. This can be expressed as:
\[ x + 14 \equiv 0 \mod 9 \]

### Step 5: Simplify the equation
To find the possible values of \( x \), we can find \( 14 \mod 9 \):
\[ 14 \div 9 = 1 \quad \text{(remainder 5)} \]
So, \( 14 \equiv 5 \mod 9 \). Therefore, we can rewrite the equation as:
\[ x + 5 \equiv 0 \mod 9 \]
This simplifies to:
\[ x \equiv -5 \mod 9 \]
Since \( -5 \) is equivalent to \( 4 \) in modulo 9 (because \( -5 + 9 = 4 \)), we have:
\[ x \equiv 4 \mod 9 \]

### Step 6: Find the value of \( x \)
The possible values of \( x \) that satisfy this equation and are single-digit numbers (0-9) is:
\[ x = 4 \]

### Step 7: Find the complete number
Now substituting \( x \) back into the number \( x806 \):
\[ 4806 \]

### Conclusion
Thus, the value of \( x \) is \( 4 \) and the complete number is \( 4806 \).

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