$3 \frac{1}{5} \times \frac{3}{N}<3 \frac{1}{5}$

Step 1 ：The question is asking for the value of N such that when \(3 \frac{1}{5}\) is multiplied by \(\frac{3}{N}\), the result is less than \(3 \frac{1}{5}\). This implies that the fraction \(\frac{3}{N}\) must be less than 1.

Step 2 ：Let's start by setting N = 4. The value of N that makes the inequality true is 4. However, since the inequality is strict (i.e., it's '<' not '<='), N must be greater than 4.

Step 3 ：Let's increment N by 1 and check again. Now, N = 5.

Step 4 ：Final Answer: The value of N that satisfies the inequality is \(\boxed{5}\).