Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

[i]Proposed by Nathan Xiong[/i]
1. **Identify the total number of outcomes:**
   - When rolling two dice, each die has 6 faces, so the total number of outcomes is:
     \[
     6 \times 6 = 36
     \]

2. **Identify non-composite products:**
   - A product is non-composite if it is either 1 or a prime number.
   - The product 1 can only occur with the pair \((1,1)\).
   - Prime products can occur with the following pairs:
     \[
     (1,2), (2,1), (1,3), (3,1), (1,5), (5,1)
     \]
   - These pairs yield the prime products 2, 3, and 5.

3. **Count the non-composite outcomes:**
   - There is 1 way to get the product 1.
   - There are 6 ways to get a prime product (as listed above).

4. **Calculate the number of composite outcomes:**
   - The total number of outcomes is 36.
   - The number of non-composite outcomes is \(1 + 6 = 7\).
   - Therefore, the number of composite outcomes is:
     \[
     36 - 7 = 29
     \]

5. **Calculate the probability of getting a composite product:**
   - The probability is the number of composite outcomes divided by the total number of outcomes:
     \[
     \frac{29}{36}
     \]

6. **Express the probability in simplest form:**
   - The fraction \(\frac{29}{36}\) is already in simplest form because 29 and 36 are relatively prime.

7. **Compute \(m+n\):**
   - Here, \(m = 29\) and \(n = 36\).
   - Therefore, \(m+n = 29 + 36 = 65\).

The final answer is \(\boxed{65}\).