Let `f_(i)(x)=sin(2p_(i)x)` for `i=1,2,3 & p_(i) in N`. If is given that the fundamental periods of `f_(1)(x)+f_(2)(x)+f_(3)(x), f_(1)(x)+f_(2)(x) and f_(1)(x)+f_(3)(x)` are `pi, (pi)/(3)` respectively, then the minimum value of `p_(1)+p_(2)+p_(3)` is

Answer

To solve the problem, we need to find the minimum value of \( p_1 + p_2 + p_3 \) given the fundamental periods of the functions.

### Step 1: Understanding the Period of Each Function
The function \( f_i(x) = \sin(2\pi p_i x) \) has a period given by:
\[ T_i = \frac{1}{p_i} \]
This means that the period of \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \) are \( \frac{1}{p_1} \), \( \frac{1}{p_2} \), and \( \frac{1}{p_3} \) respectively.

### Step 2: Setting Up the Given Information
We are given the following conditions:
1. The period of \( f_1(x) + f_2(x) + f_3(x) \) is \( \pi \).
2. The period of \( f_1(x) + f_2(x) \) is \( \frac{\pi}{3} \).
3. The period of \( f_1(x) + f_3(x) \) is \( \frac{\pi}{2} \).

### Step 3: Finding the LCM for Each Case
1. For \( f_1 + f_2 + f_3 \):
\[ \text{LCM}\left(\frac{1}{p_1}, \frac{1}{p_2}, \frac{1}{p_3}\right) = \pi \]
This implies:
\[ \frac{1}{\text{GCD}(p_1, p_2, p_3)} = \pi \implies \text{GCD}(p_1, p_2, p_3) = 1 \]

2. For \( f_1 + f_2 \):
\[ \text{LCM}\left(\frac{1}{p_1}, \frac{1}{p_2}\right) = \frac{\pi}{3} \]
This implies:
\[ \frac{1}{\text{GCD}(p_1, p_2)} = \frac{\pi}{3} \implies \text{GCD}(p_1, p_2) = 3 \]

3. For \( f_1 + f_3 \):
\[ \text{LCM}\left(\frac{1}{p_1}, \frac{1}{p_3}\right) = \frac{\pi}{2} \]
This implies:
\[ \frac{1}{\text{GCD}(p_1, p_3)} = \frac{\pi}{2} \implies \text{GCD}(p_1, p_3) = 2 \]

### Step 4: Analyzing the GCD Conditions
From the GCD conditions, we have:
- \( \text{GCD}(p_1, p_2, p_3) = 1 \)
- \( \text{GCD}(p_1, p_2) = 3 \)
- \( \text{GCD}(p_1, p_3) = 2 \)

This implies:
- \( p_1 \) must be a multiple of both 3 and 2. Hence, \( p_1 \) must be at least 6.
- \( p_2 \) must be a multiple of 3.
- \( p_3 \) must be a multiple of 2.

### Step 5: Assigning Minimum Values
Let’s assign the minimum values based on the above deductions:
- \( p_1 = 6 \)
- \( p_2 = 3 \)
- \( p_3 = 2 \)

### Step 6: Calculating the Minimum Value
Now, we can calculate:
\[ p_1 + p_2 + p_3 = 6 + 3 + 2 = 11 \]

### Final Answer
The minimum value of \( p_1 + p_2 + p_3 \) is \( \boxed{11} \).