x and y are two positive integers such that the least prime factor of x is 3 and the least prime factor of y is 5, then the least prime factor of (x + y) is :

Q: x and y are two positive integers such that the least prime factor of x is 3 and the least prime factor of y is 5, then the least prime factor of (x + y) is :

To determine the least prime factor of $x + y$, where $x$ and $y$ are positive integers with the least prime factor of $x$ being 3 and the least prime factor of $y$ being 5, we need to consider the properties of prime numbers and the sum of two numbers. ### Explanation Supporting Option 4: 1. **Understanding the Problem:** - $x$ is a positive integer whose smallest prime factor is 3. This means $x$ could be 3, 6, 9, 12, etc., but not any number that has a smaller prime factor like 2. - $y$ is a positive integer whose smallest prime factor is 5. This means $y$ could be 5, 10, 15, 20, etc., but not any number that has a smaller prime factor like 2 or 3. 2. **Analyzing $x + y$:** - Since $x$ is not divisible by 2 (as its smallest prime factor is 3), $x$ is an odd number. - Similarly, since $y$ is not divisible by 2 (as its smallest prime factor is 5), $y$ is also an odd number. - The sum of two odd numbers is always even. Therefore, $x + y$ is an even number. 3. **Determining the Least Prime Factor of an Even Number:** - The smallest prime factor of any even number is 2. This is because 2 is the smallest prime number and any even number is divisible by 2. 4. **Conclusion:** - Since $x + y$ is even, the least prime factor of $x + y$ is 2. ### Explanation for Why Other Options are Incorrect: - **Option 1 (1):** - 1 is not a prime number. Prime numbers are defined as numbers greater than 1 that have no divisors other than 1 and themselves. - **Option 2 (11):** - While 11 is a prime number, it cannot be the least prime factor of $x + y$ because $x + y$ is even, and the smallest prime factor of any even number is 2. - **Option 3 (7):** - Similar to 11, 7 is a prime number, but it cannot be the least prime factor of $x + y$ because $x + y$ is even, and the smallest prime factor of any even number is 2. Thus, the least prime factor of $x + y$ is indeed 2, making Option 4 the correct answer.

Question

Accepted Answer

To determine the least prime factor of $x + y$, where $x$ and $y$ are positive integers with the least prime factor of $x$ being 3 and the least prime factor of $y$ being 5, we need to consider the properties of prime numbers and the sum of two numbers.

### Explanation Supporting Option 4:

1. **Understanding the Problem:**
   - $x$ is a positive integer whose smallest prime factor is 3. This means $x$ could be 3, 6, 9, 12, etc., but not any number that has a smaller prime factor like 2.
   - $y$ is a positive integer whose smallest prime factor is 5. This means $y$ could be 5, 10, 15, 20, etc., but not any number that has a smaller prime factor like 2 or 3.

2. **Analyzing $x + y$:**
   - Since $x$ is not divisible by 2 (as its smallest prime factor is 3), $x$ is an odd number.
   - Similarly, since $y$ is not divisible by 2 (as its smallest prime factor is 5), $y$ is also an odd number.
   - The sum of two odd numbers is always even. Therefore, $x + y$ is an even number.

3. **Determining the Least Prime Factor of an Even Number:**
   - The smallest prime factor of any even number is 2. This is because 2 is the smallest prime number and any even number is divisible by 2.

4. **Conclusion:**
   - Since $x + y$ is even, the least prime factor of $x + y$ is 2.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (1):** 
  - 1 is not a prime number. Prime numbers are defined as numbers greater than 1 that have no divisors other than 1 and themselves.

- **Option 2 (11):**
  - While 11 is a prime number, it cannot be the least prime factor of $x + y$ because $x + y$ is even, and the smallest prime factor of any even number is 2.

- **Option 3 (7):**
  - Similar to 11, 7 is a prime number, but it cannot be the least prime factor of $x + y$ because $x + y$ is even, and the smallest prime factor of any even number is 2.

Thus, the least prime factor of $x + y$ is indeed 2, making Option 4 the correct answer.