Let 𝑎 and 𝑏 be natural numbers such that 𝑎 > 𝑏, 𝑎 + 𝑏 is divisible by 9, and 𝑎 − 𝑏 is divisible by 11.
a) Determine the smallest possible value of the number 𝑎 + 𝑏.



We are given two natural numbers \(a\) and \(b\) such that:

1. \(a > b\),
2. \(a + b\) is divisible by 9, i.e., \(a + b \equiv 0 \pmod{9}\),
3. \(a - b\) is divisible by 11, i.e., \(a - b \equiv 0 \pmod{11}\).

We need to determine the smallest possible value of \(a + b\).

### Step 1: Use the given conditions
Let’s denote:
\[
a + b = 9k \quad \text{for some integer } k,
\]
\[
a - b = 11m \quad \text{for some integer } m.
\]

From these two equations, we can solve for \(a\) and \(b\) by adding and subtracting:

\[
a + b = 9k,
\]
\[
a - b = 11m.
\]

### Step 2: Solve for \(a\) and \(b\)
Add the two equations:
\[
(a + b) + (a - b) = 9k + 11m,
\]
\[
2a = 9k + 11m,
\]
\[
a = \frac{9k + 11m}{2}.
\]

Now, subtract the second equation from the first:
\[
(a + b) - (a - b) = 9k - 11m,
\]
\[
2b = 9k - 11m,
\]
\[
b = \frac{9k - 11m}{2}.
\]

### Step 3: Ensure \(a\) and \(b\) are natural numbers
For both \(a\) and \(b\) to be natural numbers, the expressions \(\frac{9k + 11m}{2}\) and \(\frac{9k - 11m}{2}\) must be integers. This implies that \(9k + 11m\) and \(9k - 11m\) must both be even.

### Step 4: Find the smallest value of \(a + b\)
We now test values of \(k\) and \(m\) to find the smallest solution that satisfies the conditions.

For \(k = 5\) and \(m = 1\):
\[
a + b = 9 \times 5 = 45.
\]
Thus, the smallest possible value of \(a + b\) is \( \boxed{45} \).

Would you like more details or have any further questions?

Here are five related questions:
1. How would you solve this problem if 𝑎 − 𝑏 were divisible by another number?
2. What is the significance of using the system of equations method here?
3. How can this approach be extended to other modular arithmetic problems?
4. How does finding integer solutions relate to divisibility rules?
5. Could we approach this problem geometrically in number theory?

**Tip:** When solving equations involving divisibility, express each term in modular form to find integer solutions efficiently.

Let 𝑎 and 𝑏 be natural numbers such that 𝑎 > 𝑏, 𝑎 + 𝑏 is divisible by 9, and 𝑎 − 𝑏 is divisible by 11. a) Determine the smallest possible value of the number 𝑎 + 𝑏.

Explore how to determine the smallest possible value of a + b for natural numbers a and b where a > b, and a + b is divisible by 9 while a - b is divisible by 11. Learn to solve this number theory problem using modular arithmetic and systems of linear equations. Suitable for high school students in Grades 10-12.

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