Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.
To solve the problem, we need to find all prime numbers \( p \) and \( q \) such that \( p \) divides \( 30q - 1 \) and \( q \) divides \( 30p - 1 \).

1. **Initial Setup:**
   Given the conditions:
   \[
   p \mid (30q - 1) \quad \text{and} \quad q \mid (30p - 1)
   \]
   This implies there exist integers \( a \) and \( b \) such that:
   \[
   30q - 1 = ap \quad \text{and} \quad 30p - 1 = bq
   \]

2. **Combining the Equations:**
   We can rewrite the equations as:
   \[
   30q = ap + 1 \quad \text{and} \quad 30p = bq + 1
   \]
   Adding these two equations, we get:
   \[
   30q + 30p = ap + bq + 2
   \]
   Rearranging, we have:
   \[
   30(p + q) = ap + bq + 2
   \]

3. **Exploring Possible Values of \( k \):**
   Let \( k = \frac{30(p + q) - 2}{pq} \). Since \( k \) must be an integer, we need to find values of \( k \) such that:
   \[
   k(pq) = 30(p + q) - 2
   \]
   Given that \( k \) must be a divisor of \( 30 \) and \( \gcd(30, k) = 1 \), the possible values of \( k \) are \( 1, 7, 11, 13, 17, 19, 23, 29 \).

4. **Analyzing Each \( k \):**
   - For \( k = 1 \):
     \[
     pq = 30(p + q) - 2
     \]
     Rearranging, we get:
     \[
     pq - 30p - 30q + 2 = 0
     \]
     Solving this quadratic equation in terms of \( p \) and \( q \), we find:
     \[
     (p - 30)(q - 30) = 899
     \]
     The factor pairs of 899 are \( (1, 899) \) and \( (29, 31) \). Since \( p \) and \( q \) are primes, the valid pairs are:
     \[
     (p, q) = (59, 61) \quad \text{and} \quad (61, 59)
     \]

   - For \( k = 7 \):
     \[
     7pq = 30(p + q) - 2
     \]
     Rearranging, we get:
     \[
     (7p - 30)(7q - 30) = 893
     \]
     The factor pairs of 893 are \( (1, 893) \) and \( (19, 47) \). Since \( p \) and \( q \) are primes, the valid pairs are:
     \[
     (p, q) = (7, 11) \quad \text{and} \quad (11, 7)
     \]

   - For \( k = 11 \):
     \[
     11pq = 30(p + q) - 2
     \]
     Rearranging, we get:
     \[
     (11p - 30)(11q - 30) = 889
     \]
     The factor pairs of 889 are \( (1, 889) \) and \( (23, 37) \). Since \( p \) and \( q \) are primes, there are no valid pairs.

5. **Conclusion:**
   The valid pairs of primes \( (p, q) \) that satisfy the given conditions are:
   \[
   (p, q) = (7, 11), (11, 7), (59, 61), (61, 59)
   \]

The final answer is \(\boxed{(7, 11), (11, 7), (59, 61), (61, 59)}\).