Find the greatest value of n such that the number 1578n is divisible by 3.

Options

A. 3
B. 9
C. 6
D. 8

Answer

To find the greatest value of \( n \) such that the number \( 1578n \) is divisible by 3, we need to follow these steps:

### Step 1: Understand the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 2: Calculate the sum of the known digits
The digits of the number \( 1578n \) are \( 1, 5, 7, 8, \) and \( n \). First, we will calculate the sum of the digits without \( n \):
\[ 1 + 5 + 7 + 8 = 21 \]

### Step 3: Add \( n \) to the sum
Now, we include \( n \) in the sum:
\[ \text{Sum} = 21 + n \]

### Step 4: Determine the values of \( n \)
We need \( 21 + n \) to be divisible by 3. Since \( 21 \) is already divisible by 3 (as \( 21 \div 3 = 7 \)), we need \( n \) to also be divisible by 3.

### Step 5: List possible values for \( n \)
The possible digits for \( n \) (since \( n \) is a single digit) are \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \). The digits that are divisible by 3 are:
- \( 0 \)
- \( 3 \)
- \( 6 \)
- \( 9 \)

### Step 6: Identify the greatest value of \( n \)
Among the possible values \( 0, 3, 6, \) and \( 9 \), the greatest value is \( 9 \).

### Conclusion
Thus, the greatest value of \( n \) such that \( 1578n \) is divisible by 3 is:
\[ \boxed{9} \]

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