Find all positive integers $n$ such that there are $\infty$ many lines of Pascal's triangle that have entries coprime to $n$ only. In other words: such that there are $\infty$ many $k$ with the property that the numbers $\binom{k}{0},\binom{k}{1},\binom{k}{2},...,\binom{k}{k}$ are all coprime to $n$.
1. **Understanding the Problem:**
   We need to find all positive integers \( n \) such that there are infinitely many rows in Pascal's triangle where all the binomial coefficients in that row are coprime to \( n \). This means for infinitely many \( k \), all the numbers \(\binom{k}{0}, \binom{k}{1}, \binom{k}{2}, \ldots, \binom{k}{k}\) are coprime to \( n \).

2. **Using Lucas' Theorem:**
   Lucas' Theorem states that for a prime \( p \), the binomial coefficient \(\binom{n}{m}\) modulo \( p \) can be determined by the base \( p \) representations of \( n \) and \( m \). Specifically, \(\binom{n}{m} \equiv \prod_{i=0}^k \binom{n_i}{m_i} \pmod{p}\), where \( n_i \) and \( m_i \) are the digits of \( n \) and \( m \) in base \( p \).

3. **Condition for Coprimality:**
   For \(\binom{k}{i}\) to be coprime to \( n \) for all \( i \) from \( 0 \) to \( k \), \( n \) must not divide any of the binomial coefficients in that row. This implies that \( n \) must not share any prime factors with any of the binomial coefficients in that row.

4. **Prime Power Analysis:**
   If \( n \) is a power of a prime \( p \), say \( n = p^e \), then by Lucas' Theorem, \(\binom{k}{i} \not\equiv 0 \pmod{p}\) for all \( i \) if and only if \( k \) is of the form \( ap^l - 1 \) for some \( l \) and \( a \) such that \( 1 \leq a < p \). This ensures that the binomial coefficients in that row are not divisible by \( p \).

5. **Infinitely Many Rows:**
   For there to be infinitely many such \( k \), \( n \) must be such that the condition holds for infinitely many \( k \). This is possible if and only if \( n \) is a power of a prime. If \( n \) were not a power of a prime, it would have multiple prime factors, and it would be impossible to find infinitely many rows where all binomial coefficients are coprime to \( n \).

6. **Conclusion:**
   Therefore, \( n \) must be a power of a prime for there to be infinitely many rows in Pascal's triangle where all binomial coefficients are coprime to \( n \).

\(\blacksquare\)

The final answer is \( \boxed{ n } \) must be a power of a prime.