Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$.  A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$.

[i]Proposed by Eugene Chen[/i]
1. **Identify the set \( S \)**:
   - \( S \) is the set of integers that are both multiples of \( 70 \) and factors of \( 630,000 \).
   - First, factorize \( 70 \) and \( 630,000 \):
     \[
     70 = 2 \times 5 \times 7
     \]
     \[
     630,000 = 2^3 \times 3^2 \times 5^3 \times 7
     \]
   - Any element \( c \) in \( S \) must be of the form \( 70k \) where \( k \) is a factor of \( \frac{630,000}{70} = 9000 \):
     \[
     9000 = 2^3 \times 3^2 \times 5^3
     \]

2. **Determine the number of elements in \( S \)**:
   - The number of factors of \( 9000 \) is given by:
     \[
     (3+1)(2+1)(3+1) = 4 \times 3 \times 4 = 48
     \]
   - Therefore, there are 48 elements in \( S \).

3. **Condition for \( \gcd(c,d) = 70 \) and \( \operatorname{lcm}(c,d) = 630,000 \)**:
   - Let \( c = 70k \) and \( d = 70j \) where \( k \) and \( j \) are coprime.
   - Given \( \gcd(c,d) = 70 \), we have \( \gcd(70k, 70j) = 70 \gcd(k,j) \). Thus, \( \gcd(k,j) = 1 \).
   - Given \( \operatorname{lcm}(c,d) = 630,000 \), we have:
     \[
     \operatorname{lcm}(70k, 70j) = 70 \times \operatorname{lcm}(k,j) = 630,000
     \]
     \[
     \operatorname{lcm}(k,j) = \frac{630,000}{70} = 9000
     \]
   - Since \( k \) and \( j \) are coprime, \( k \times j = 9000 \).

4. **Count the number of valid pairs \((k, j)\)**:
   - The prime factorization of \( 9000 \) is \( 2^3 \times 3^2 \times 5^3 \).
   - Each prime factor must be entirely in either \( k \) or \( j \) since \( k \) and \( j \) are coprime.
   - The number of ways to distribute the prime factors between \( k \) and \( j \) is:
     \[
     2^3 = 8
     \]
   - Therefore, there are 8 valid pairs \((k, j)\).

5. **Calculate the probability**:
   - The probability that a randomly selected \( c \) from \( S \) satisfies the conditions is:
     \[
     \frac{8}{48} = \frac{1}{6}
     \]
   - Here, \( m = 1 \) and \( n = 6 \).

6. **Compute \( 100m + n \)**:
   - Substituting \( m \) and \( n \):
     \[
     100m + n = 100 \times 1 + 6 = 106
     \]

The final answer is \( \boxed{106} \)