A natural number $n$ is called breakable if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$. Find all breakable numbers.
1. **Suppose \( n \) is a composite number.**
   - By definition, a composite number \( n \) can be written as \( n = st \) where \( s \) and \( t \) are natural numbers greater than 1.
   - Let \( a = (s-1)t \), \( b = t \), \( x = (s-1)(t-1) \), and \( y = 1 \).
   - We need to verify that \( a + b = n \) and \(\frac{x}{a} + \frac{y}{b} = 1\).

2. **Check the sum \( a + b \):**
   \[
   a + b = (s-1)t + t = st = n
   \]
   - This confirms that \( a + b = n \).

3. **Check the equation \(\frac{x}{a} + \frac{y}{b} = 1\):**
   \[
   \frac{x}{a} + \frac{y}{b} = \frac{(s-1)(t-1)}{(s-1)t} + \frac{1}{t}
   \]
   - Simplify the first term:
   \[
   \frac{(s-1)(t-1)}{(s-1)t} = \frac{t-1}{t}
   \]
   - Now, add the two fractions:
   \[
   \frac{t-1}{t} + \frac{1}{t} = \frac{t-1+1}{t} = \frac{t}{t} = 1
   \]
   - This confirms that \(\frac{x}{a} + \frac{y}{b} = 1\).

4. **Therefore, if \( n \) is composite, it is breakable.**

5. **Suppose \( n \) is a prime number.**
   - Assume \( n \) is breakable, so there exist positive integers \( a, b, x, y \) such that \( a + b = n \) and \(\frac{x}{a} + \frac{y}{b} = 1\).
   - Since \( n \) is prime, \( a \) and \( b \) must be coprime (i.e., \(\gcd(a, b) = 1\)).

6. **From \(\frac{x}{a} + \frac{y}{b} = 1\):**
   \[
   \frac{x}{a} + \frac{y}{b} = 1 \implies bx + ay = ab
   \]
   - Rearrange to get:
   \[
   bx = a(b - y)
   \]
   - Since \( \gcd(a, b) = 1 \), \( a \) must divide \( x \).

7. **Let \( x = ka \) for some integer \( k \):**
   \[
   \frac{ka}{a} + \frac{y}{b} = 1 \implies k + \frac{y}{b} = 1 \implies \frac{y}{b} = 1 - k
   \]
   - Since \( y \) and \( b \) are positive integers, \( 1 - k \) must be a positive rational number, implying \( k < 1 \).

8. **Since \( k \) is an integer, \( k \leq 0 \):**
   - But \( k \) must be positive because \( x = ka \) and \( x \) is positive.
   - This contradiction implies that \( n \) cannot be breakable if \( n \) is prime.

9. **Therefore, if \( n \) is prime, it is not breakable.**

The final answer is \( \boxed{ n } \) is breakable if and only if \( n \) is a composite number.