Which one of the following is wrong?

Options

A. The sum of two even numbers, each raised to an odd power is even
B. The sum of two odd numbers, each raised to an odd power is even
C. The remainder when dividing an even number by an odd number is even or zero
D. The remainder when dividing an odd number by an even number is always odd

Answer

To determine which of the statements is wrong, we will analyze each statement one by one.

### Step 1: Analyze the first statement
**Statement:** The sum of two even numbers each raised to an odd power is even.

**Example:**
- Let’s take two even numbers: 2 and 4.
- Calculate: \(2^3 + 4^3\)
- \(2^3 = 8\) and \(4^3 = 64\)
- Now, sum them: \(8 + 64 = 72\)

**Conclusion:** The sum is even. This statement is correct.

### Step 2: Analyze the second statement
**Statement:** The sum of two odd numbers each raised to an odd power is even.

**Example:**
- Let’s take two odd numbers: 3 and 5.
- Calculate: \(3^3 + 5^3\)
- \(3^3 = 27\) and \(5^3 = 125\)
- Now, sum them: \(27 + 125 = 152\)

**Conclusion:** The sum is even. This statement is correct.

### Step 3: Analyze the third statement
**Statement:** The remainder when dividing an even number by an odd number is even or zero.

**Example:**
- Let’s take an even number: 6 and divide it by an odd number: 3.
- \(6 \div 3 = 2\) with a remainder of 0.
- Now, let’s take another example: \(16 \div 5\).
- \(16 = 5 \times 3 + 1\) (remainder is 1).
- Another example: \(24 \div 7\).
- \(24 = 7 \times 3 + 3\) (remainder is 3).

**Conclusion:** The remainders we are getting are odd (1, 3) or zero, not even. This statement is incorrect.

### Step 4: Analyze the fourth statement
**Statement:** The remainder when dividing an odd number by an even number is always odd.

**Example:**
- Let’s take an odd number: 7 divided by an even number: 2.
- \(7 \div 2 = 3\) with a remainder of 1.
- Another example: \(17 \div 4\).
- \(17 = 4 \times 4 + 1\) (remainder is 1).
- Another example: \(65 \div 8\).
- \(65 = 8 \times 8 + 1\) (remainder is 1).

**Conclusion:** The remainders are consistently odd (1). This statement is correct.

### Final Conclusion
The incorrect statement is the **third statement**: "The remainder when dividing an even number by an odd number is even or zero."

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