Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$?

[i]Proposed by Oleksiy Masalitin[/i]
1. **Assume $M$ contains $n$ positive integers $a_1, a_2, \ldots, a_n$.** 
   We are given that for any two different $a, b \in M$, $a^2 + 1$ is divisible by $b$. This implies that for any $a_i, a_j \in M$ with $i \neq j$, $a_i^2 + 1$ is divisible by $a_j$.

2. **Consider the smallest element $a_1$ in $M$.** 
   Without loss of generality, let $a_1$ be the smallest element in $M$. Then, for each $a_i \in M$ where $i \neq 1$, $a_i$ must divide $a_1^2 + 1$.

3. **Analyze the divisibility condition.**
   Since $a_i$ divides $a_1^2 + 1$ for all $i \neq 1$, we have:
   \[
   a_2 \cdot a_3 \cdot \ldots \cdot a_n \leq a_1^2 + 1
   \]
   This is because each $a_i$ is a divisor of $a_1^2 + 1$ and they are all distinct positive integers.

4. **Consider the case when $n \geq 3$.**
   If $n \geq 3$, then the product of $a_2, a_3, \ldots, a_n$ must be greater than $a_1^2 + 1$. Specifically, since $a_1$ is the smallest element, we have:
   \[
   a_2 \cdot a_3 \cdot \ldots \cdot a_n \geq (a_1 + 1)^{n-1}
   \]
   For $n \geq 3$, this product will be greater than $a_1^2 + 1$:
   \[
   (a_1 + 1)^{n-1} > a_1^2 + 1
   \]
   This inequality holds because $(a_1 + 1)^{n-1}$ grows faster than $a_1^2 + 1$ as $n$ increases.

5. **Conclude that $n \geq 3$ is not possible.**
   Since the product of $a_2, a_3, \ldots, a_n$ exceeds $a_1^2 + 1$ for $n \geq 3$, it contradicts the given condition that $a_i$ divides $a_1^2 + 1$. Therefore, $n \geq 3$ is not possible.

6. **Verify that $n = 2$ works.**
   For $n = 2$, consider the set $M = \{2, 5\}$. We check the condition:
   \[
   2^2 + 1 = 5 \quad \text{(divisible by 5)}
   \]
   \[
   5^2 + 1 = 26 \quad \text{(divisible by 2)}
   \]
   Both conditions are satisfied, so $n = 2$ is possible.

7. **Conclude the largest possible value of $n$.**
   Since $n \geq 3$ fails and $n = 2$ works, the largest possible value of $n$ is 2.

The final answer is $\boxed{2}$