Higher Secondary P5

Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
1. **Assume \( x \) is composite:**
   - Let \( x \) be a composite number with more than one prime factor.
   - We can write \( x \) as \( x = p_1^{a_1} p_2^{a_2} \cdots p_j^{a_j} \cdots p_n^{a_n} \), where \( a_i \ge 1 \) and all \( p_i \) are distinct primes, with \( 1 \le j < n \).

2. **Construct \( a \) and \( b \):**
   - Let \( a = p_1^{a_1} p_2^{a_2} \cdots p_j^{a_j} \) and \( b = p_j^{a_j} \cdots p_n^{a_n} \).
   - Then \( ab = x \), so \( x \) divides \( ab \).

3. **Check divisibility:**
   - By the given condition, if \( x \) divides \( ab \), then \( x \) must divide either \( a \) or \( b \).
   - However, \( x \) does not divide \( a \) because \( a \) is missing some prime factors of \( x \).
   - Similarly, \( x \) does not divide \( b \) because \( b \) is also missing some prime factors of \( x \).
   - This leads to a contradiction.

4. **Conclude \( x \) is not composite:**
   - Since assuming \( x \) is composite leads to a contradiction, \( x \) must be a power of a single prime.

5. **Assume \( x = p^a \) where \( p \) is a prime and \( a > 1 \):**
   - Let \( a = p \) and \( b = p^{a-1} \).
   - Then \( ab = p \cdot p^{a-1} = p^a = x \), so \( x \) divides \( ab \).

6. **Check divisibility for \( x = p^a \):**
   - By the given condition, if \( x \) divides \( ab \), then \( x \) must divide either \( a \) or \( b \).
   - However, \( x \) does not divide \( a = p \) because \( p \) is not equal to \( p^a \).
   - Similarly, \( x \) does not divide \( b = p^{a-1} \) because \( p^{a-1} \) is not equal to \( p^a \).
   - This leads to a contradiction.

7. **Conclude \( x \) must be a prime:**
   - Since assuming \( x = p^a \) with \( a > 1 \) leads to a contradiction, \( x \) must be a prime number.
   - Therefore, \( x = p \) for some prime \( p \).

8. **Determine the number of positive divisors of \( x \):**
   - Since \( x \) is a prime number, it has exactly two positive divisors: \( 1 \) and \( x \).

The final answer is \( \boxed{ 2 } \).