Given the integer $n\geq 2$ and a integer ${a}$, which is coprime with ${n}$. A country has ${n}$ islands $D_1$, $D_2$, $\cdots$, $D_n$. For any $1\leq i\neq j\leq n$, there is a one-way ferry $D_i$ to $D_j$ if and only if $ij\equiv ia\pmod n$. A tourist can initially fly to any of the islands, and then he can only take a one-way ferry. What is the maximum number of islands he can visit?

[i]Created by Zhenhua Qu[/i]
To solve this problem, we need to understand the structure of the islands and the ferry connections. The key is to analyze the condition for the existence of a ferry between two islands \(D_i\) and \(D_j\).

1. **Condition for Ferry Connection**:
   The problem states that there is a one-way ferry from \(D_i\) to \(D_j\) if and only if \(ij \equiv ia \pmod{n}\). This can be simplified as:
   \[
   ij \equiv ia \pmod{n} \implies j \equiv a \pmod{n}
   \]
   This means that for any island \(D_i\), there is a ferry to another island \(D_j\) if and only if \(j \equiv a \pmod{n}\).

2. **Coprime Condition**:
   Since \(a\) is coprime with \(n\), \(a\) has an inverse modulo \(n\). This implies that the mapping \(i \mapsto ai \pmod{n}\) is a permutation of the set \(\{1, 2, \ldots, n\}\).

3. **Cycle Structure**:
   The permutation induced by \(i \mapsto ai \pmod{n}\) can be decomposed into disjoint cycles. Each cycle represents a sequence of islands that can be visited starting from any island in that cycle.

4. **Maximum Number of Islands**:
   The maximum number of islands a tourist can visit corresponds to the length of the longest cycle in the permutation. The length of the longest cycle is determined by the order of \(a\) modulo \(n\), which is the smallest positive integer \(k\) such that \(a^k \equiv 1 \pmod{n}\).

5. **Function \(\gamma(n)\)**:
   The function \(\gamma(n)\) is defined based on the prime factorization of \(n\). Specifically, if \(n\) has the form \(n = 2^{\alpha} p_1 p_2 \cdots p_i p_{i+1}^{\alpha_{i+1}} \cdots p_m^{\alpha_m}\) with \(\alpha_{i+1}, \ldots, \alpha_m \geq 2\), then:
   \[
   \gamma(n) = \begin{cases}
   3m - i & \text{if } \alpha = 0 \\
   3m - i + 1 & \text{if } \alpha = 1 \\
   3m - i + 3 & \text{if } \alpha \geq 2
   \end{cases}
   \]
   This function \(\gamma(n)\) essentially counts the number of distinct prime factors and their exponents to determine the structure of the cycles.

6. **Conclusion**:
   The maximum number of islands a tourist can visit is given by \(1 + \gamma(n)\).

The final answer is \(\boxed{1 + \gamma(n)}\).