Find the least possible value of a + b, where a, b are positive integers such that 11 divides a + 13b and 13 divides a + 11b.

Answer

To find the least possible value of \( a + b \) where \( a \) and \( b \) are positive integers such that \( 11 \) divides \( a + 13b \) and \( 13 \) divides \( a + 11b \), we can follow these steps:

### Step 1: Set Up the Conditions
We start with the two conditions given in the problem:
1. \( a + 13b \equiv 0 \mod{11} \)
2. \( a + 11b \equiv 0 \mod{13} \)

### Step 2: Rewrite the Conditions
From the first condition:
\[ a + 13b \equiv 0 \mod{11} \implies a \equiv -13b \mod{11} \]
Since \( 13 \equiv 2 \mod{11} \), we can rewrite this as:
\[ a \equiv -2b \mod{11} \implies a \equiv 9b \mod{11} \]

From the second condition:
\[ a + 11b \equiv 0 \mod{13} \implies a \equiv -11b \mod{13} \]
Since \( 11 \equiv -2 \mod{13} \), we can rewrite this as:
\[ a \equiv 2b \mod{13} \]

### Step 3: Set Up a System of Congruences
Now we have a system of two congruences:
1. \( a \equiv 9b \mod{11} \)
2. \( a \equiv 2b \mod{13} \)

### Step 4: Solve the System of Congruences
We can use the method of substitution. From the first congruence, we can express \( a \) in terms of \( b \):
\[ a = 9b + 11k \quad \text{for some integer } k \]
Substituting this into the second congruence:
\[ 9b + 11k \equiv 2b \mod{13} \]
This simplifies to:
\[ 7b + 11k \equiv 0 \mod{13} \]
Now, we can express \( 11k \mod{13} \):
\[ 11 \equiv -2 \mod{13} \implies -2k \equiv -7b \mod{13} \implies 2k \equiv 7b \mod{13} \]
Multiplying both sides by the modular inverse of \( 2 \mod{13} \) (which is \( 7 \)):
\[ k \equiv 7 \cdot 7b \mod{13} \implies k \equiv 49b \mod{13} \implies k \equiv 10b \mod{13} \]

### Step 5: Substitute Back to Find \( a \)
Now substituting \( k \) back into the expression for \( a \):
\[ k = 10b + 13m \quad \text{for some integer } m \]
Substituting this into \( a \):
\[ a = 9b + 11(10b + 13m) = 9b + 110b + 143m = 119b + 143m \]

### Step 6: Find \( a + b \)
Now we want to minimize \( a + b \):
\[ a + b = 119b + 143m + b = 120b + 143m \]

### Step 7: Minimize \( a + b \)
To minimize \( 120b + 143m \), we can try small values for \( b \) and \( m \):
- Let \( m = 0 \) and try small values of \( b \):
- For \( b = 1 \): \( a + b = 120(1) + 143(0) = 120 \)
- For \( b = 2 \): \( a + b = 120(2) + 143(0) = 240 \)
- For \( b = 3 \): \( a + b = 120(3) + 143(0) = 360 \)

Continuing this way, we find that the least possible value occurs when \( b = 1 \) and \( m = 0 \), giving:
\[ a + b = 120 \]

### Conclusion
Thus, the least possible value of \( a + b \) is \( 120 \).