To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) such that:
1. \( 373a \) is divisible by 9
2. \( 473b \) is divisible by 11
3. \( 371c \) is divisible by 6
Let's solve this step by step.
### Step 1: Finding \( a \)
To determine \( a \), we need \( 373a \) to be divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9.
The digits of \( 373a \) are \( 3, 7, 3, \) and \( a \).
Calculating the sum of the digits:
\[
3 + 7 + 3 + a = 13 + a
\]
Now, we need \( 13 + a \) to be divisible by 9. The possible values for \( 13 + a \) can be:
- 9
- 18
- 27
Since \( a \) must be a single digit (0-9), we can only consider:
1. \( 13 + a = 18 \) → \( a = 5 \)
Thus, \( a = 5 \).
### Step 2: Finding \( b \)
Next, we need \( 473b \) to be divisible by 11. A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or divisible by 11.
The digits of \( 473b \) are \( 4, 7, 3, \) and \( b \).
Calculating the sums:
- Odd positions: \( 4 + 3 = 7 \)
- Even positions: \( 7 + b \)
Now, we need to find:
\[
|7 - (7 + b)| = |7 - 7 - b| = |-b| = b
\]
For \( 473b \) to be divisible by 11, \( b \) must equal 0 (since it cannot be negative). Thus, \( b = 0 \).
### Step 3: Finding \( c \)
Now, we need \( 371c \) to be divisible by 6. A number is divisible by 6 if it is divisible by both 2 and 3.
**Divisibility by 2:**
For \( 371c \) to be divisible by 2, \( c \) must be even. The possible values for \( c \) are \( 0, 2, 4, 6, 8 \).
**Divisibility by 3:**
Now, we check the sum of the digits:
\[
3 + 7 + 1 + c = 11 + c
\]
We need \( 11 + c \) to be divisible by 3.
Testing the even values of \( c \):
- If \( c = 0 \): \( 11 + 0 = 11 \) (not divisible by 3)
- If \( c = 2 \): \( 11 + 2 = 13 \) (not divisible by 3)
- If \( c = 4 \): \( 11 + 4 = 15 \) (divisible by 3)
- If \( c = 6 \): \( 11 + 6 = 17 \) (not divisible by 3)
- If \( c = 8 \): \( 11 + 8 = 19 \) (not divisible by 3)
Thus, the only valid value for \( c \) is \( 4 \).
### Final Calculation
Now we have:
- \( a = 5 \)
- \( b = 0 \)
- \( c = 4 \)
Now, we can find \( a + b + c \):
\[
a + b + c = 5 + 0 + 4 = 9
\]
### Conclusion
The value of \( a + b + c \) is \( 9 \).
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