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 off 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 statement is wrong, we will analyze each of the provided options step by step.

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

- Let’s take two even numbers, for example, 2 and 4.
- Raise them to an odd power: \(2^3\) and \(4^1\).
- Calculate:
\[ 2^3 = 8 \quad \text{and} \quad 4^1 = 4 \]
- Now, find the sum:
\[ 8 + 4 = 12 \]
- Since 12 is even, **Statement 1 is correct**.

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

- Let’s take two odd numbers, for example, 1 and 3.
- Raise them to an odd power: \(1^1\) and \(3^1\).
- Calculate:
\[ 1^1 = 1 \quad \text{and} \quad 3^1 = 3 \]
- Now, find the sum:
\[ 1 + 3 = 4 \]
- Since 4 is even, **Statement 2 is correct**.

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

- Let’s take an even number, for example, 10, and divide it by an odd number, say 5.
- Perform the division:
\[ 10 \div 5 = 2 \quad \text{with a remainder of } 0 \]
- Now, let’s take another even number, 12, and divide it by 5:
\[ 12 \div 5 = 2 \quad \text{with a remainder of } 2 \]
- Both remainders (0 and 2) are even, so **Statement 3 is correct**.

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

- Let’s take an odd number, for example, 7, and divide it by an even number, say 2.
- Perform the division:
\[ 7 \div 2 = 3 \quad \text{with a remainder of } 1 \]
- Now, let’s take another odd number, 9, and divide it by 4:
\[ 9 \div 4 = 2 \quad \text{with a remainder of } 1 \]
- However, if we take 13 and divide it by 4:
\[ 13 \div 4 = 3 \quad \text{with a remainder of } 1 \]
- But if we take 17 and divide it by 6:
\[ 17 \div 6 = 2 \quad \text{with a remainder of } 5 \]
- The remainder can be odd or even depending on the numbers chosen. Therefore, **Statement 4 is incorrect**.

### Conclusion
The wrong statement is **Statement 4**.

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