When a is divided by 7 ,the remainder is 4. when b is divided by 3, the remainder is 2. If `oltalt24 and 2ltblt8`, which of the following could have remainder of 0 when divided by 8?

Options

A. `(a)/(b)`
B. `(b)/(a)`
C. `a+b`
D. `ab`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) based on the given conditions and then check which combinations of \( a \) and \( b \) yield a sum that is divisible by 8 (i.e., has a remainder of 0 when divided by 8).

### Step 1: Determine possible values for \( a \)
The problem states that when \( a \) is divided by 7, the remainder is 4. This can be expressed mathematically as:
\[ a \equiv 4 \ (\text{mod} \ 7) \]
This means that \( a \) can be expressed in the form:
\[ a = 7k + 4 \]
for some integer \( k \).

Given that \( a \) is between 0 and 24, we can find the possible values of \( a \):
- For \( k = 0 \): \( a = 7(0) + 4 = 4 \)
- For \( k = 1 \): \( a = 7(1) + 4 = 11 \)
- For \( k = 2 \): \( a = 7(2) + 4 = 18 \)
- For \( k = 3 \): \( a = 7(3) + 4 = 25 \) (not valid since it exceeds 24)

Thus, the possible values of \( a \) are:
\[ a = 4, 11, 18 \]

### Step 2: Determine possible values for \( b \)
The problem states that when \( b \) is divided by 3, the remainder is 2. This can be expressed as:
\[ b \equiv 2 \ (\text{mod} \ 3) \]
This means that \( b \) can be expressed in the form:
\[ b = 3m + 2 \]
for some integer \( m \).

Given that \( b \) is between 2 and 8, we can find the possible values of \( b \):
- For \( m = 0 \): \( b = 3(0) + 2 = 2 \)
- For \( m = 1 \): \( b = 3(1) + 2 = 5 \)
- For \( m = 2 \): \( b = 3(2) + 2 = 8 \)

Thus, the possible values of \( b \) are:
\[ b = 2, 5, 8 \]

### Step 3: Find combinations of \( a \) and \( b \)
Now we will check the combinations of \( a \) and \( b \) to see which sums are divisible by 8.

1. **For \( a = 4 \)**:
- \( b = 2 \): \( 4 + 2 = 6 \) (not divisible by 8)
- \( b = 5 \): \( 4 + 5 = 9 \) (not divisible by 8)
- \( b = 8 \): \( 4 + 8 = 12 \) (not divisible by 8)

2. **For \( a = 11 \)**:
- \( b = 2 \): \( 11 + 2 = 13 \) (not divisible by 8)
- \( b = 5 \): \( 11 + 5 = 16 \) (divisible by 8)
- \( b = 8 \): \( 11 + 8 = 19 \) (not divisible by 8)

3. **For \( a = 18 \)**:
- \( b = 2 \): \( 18 + 2 = 20 \) (not divisible by 8)
- \( b = 5 \): \( 18 + 5 = 23 \) (not divisible by 8)
- \( b = 8 \): \( 18 + 8 = 26 \) (not divisible by 8)

### Conclusion
The only combination that yields a sum divisible by 8 is:
- \( a = 11 \) and \( b = 5 \) which gives \( a + b = 16 \).

Thus, the answer is:
\[ a + b = 16 \]