p and q are two positive integers such that the least prime factor of p is 3 and the least prime factor of q is 5 . Find the least prime factor of (p + q).

Answer

To solve the problem step-by-step, we need to find the least prime factor of \( p + q \), given that the least prime factor of \( p \) is 3 and the least prime factor of \( q \) is 5.

### Step 1: Understanding the conditions for \( p \) and \( q \)
Since the least prime factor of \( p \) is 3, \( p \) must be an odd integer (because all even integers have 2 as their least prime factor). The smallest possible value for \( p \) that meets this condition is 3 itself. However, \( p \) can also be any odd integer greater than or equal to 3.

Similarly, since the least prime factor of \( q \) is 5, \( q \) must also be an odd integer. The smallest possible value for \( q \) that meets this condition is 5, but \( q \) can also be any odd integer greater than or equal to 5.

### Step 2: Choosing values for \( p \) and \( q \)
Let's choose the smallest values for \( p \) and \( q \) to simplify our calculations:
- Let \( p = 3 \)
- Let \( q = 5 \)

### Step 3: Calculate \( p + q \)
Now, we can calculate \( p + q \):
\[ p + q = 3 + 5 = 8 \]

### Step 4: Finding the least prime factor of \( p + q \)
Next, we need to find the least prime factor of 8. The prime factorization of 8 is:
\[ 8 = 2 \times 2 \times 2 \]
The least prime factor of 8 is 2.

### Conclusion
Thus, the least prime factor of \( p + q \) is:
\[ \text{Least prime factor of } (p + q) = 2 \]