If a and b are positive integers such that the remainder is 4 when a is divided by b, what is the smallest possible value of `a+b`?

Answer

To solve the problem, we need to find the smallest possible value of \( a + b \) given that when \( a \) is divided by \( b \), the remainder is 4. Here’s a step-by-step solution:

### Step 1: Understand the condition
We know that when \( a \) is divided by \( b \), the remainder is 4. This can be expressed mathematically as:
\[ a = kb + 4 \]
for some integer \( k \). This implies that \( a \) must be greater than 4 since the remainder cannot be equal to or greater than the divisor.

### Step 2: Determine the constraints on \( b \)
Since the remainder is 4, it follows that \( b \) must be greater than 4. If \( b \) were 4 or less, it would not be possible to have a remainder of 4. Therefore, the smallest integer value \( b \) can take is 5.

### Step 3: Find the minimum value of \( a \)
Now, we set \( b = 5 \) (the smallest possible value for \( b \)). We can substitute this back into our equation:
\[ a = kb + 4 \]
For the smallest possible value of \( a \), we can set \( k = 0 \) (the smallest non-negative integer):
\[ a = 0 \cdot 5 + 4 = 4 \]

### Step 4: Calculate \( a + b \)
Now we have:
- \( a = 4 \)
- \( b = 5 \)

Thus, we can calculate:
\[ a + b = 4 + 5 = 9 \]

### Conclusion
The smallest possible value of \( a + b \) is \( 9 \).