If x and y are positive integers and (2x +y) is even, which of the following must be even ?

Options

A. `x^(3) + 2xy^(2)`
B. `3x^(2) + 2y`
C. `4x^(2) + x + y^(2)`
D. `5x^(2) + x + y^(3)`

Answer

To solve the problem, we need to analyze the given condition that \(2x + y\) is even, where \(x\) and \(y\) are positive integers.

### Step-by-Step Solution:

1. **Understanding the Condition**:
- The expression \(2x + y\) is even.
- Since \(2x\) is always even (as it is a multiple of 2), for the sum \(2x + y\) to be even, \(y\) must also be even.
- Therefore, we conclude that \(y\) is an even integer.

2. **Analyzing the Options**:
- We need to check which of the given options must be even.

3. **Option 1: \(x^3 + 2xy^2\)**:
- The term \(2xy^2\) is even because it is a multiple of 2 (since \(y\) is even).
- The term \(x^3\) can be either even or odd depending on whether \(x\) is even or odd.
- Therefore, \(x^3 + 2xy^2\) can be odd if \(x\) is odd, so this option is not guaranteed to be even.

4. **Option 2: \(3x^2 + 2y\)**:
- The term \(2y\) is even (since \(y\) is even).
- The term \(3x^2\) can also be odd or even depending on \(x\).
- Hence, \(3x^2 + 2y\) can be odd if \(x\) is odd, so this option is not guaranteed to be even.

5. **Option 3: \(4x^2 + x + y^2\)**:
- The term \(4x^2\) is even (as it is a multiple of 4).
- The term \(y^2\) is even (since \(y\) is even).
- The term \(x\) can be odd or even.
- Thus, \(4x^2 + x + y^2\) can be odd if \(x\) is odd, so this option is not guaranteed to be even.

6. **Option 4: \(5x^2 + x + y^3\)**:
- The term \(y^3\) is even (since \(y\) is even).
- The term \(5x^2\) can be odd or even depending on \(x\).
- The term \(x\) can also be odd or even.
- If \(x\) is odd, both \(5x^2\) and \(x\) are odd, and the sum of two odd numbers is even. Adding the even \(y^3\) keeps the total even.
- If \(x\) is even, \(5x^2\) and \(x\) are both even, and adding the even \(y^3\) keeps the total even.
- Therefore, \(5x^2 + x + y^3\) is guaranteed to be even.

### Conclusion:
The only option that must be even is **Option 4: \(5x^2 + x + y^3\)**.