Question

Which of the following statement is incorrect ?

• A. Even number x even number = even number
• B. Even number x odd number = even number
• C. Odd number x odd number = odd number
• D. Odd number x even number = odd number

Hint:

If you ever forget multiplication parity rules during an exam, simply plug in basic test values like 2 for even numbers and 3 for odd numbers to verify the properties instantly!

Answer 1

The Correct Option is D

Concept: This question tests basic properties of multiplication operations involving even and odd integers. By definition:

• An even number is any integer that can be represented in the general algebraic form \(2n\), where \(n\) is an integer. It is perfectly divisible by 2.

• An odd number is any integer that can be represented in the general algebraic form \(2k + 1\), where \(k\) is an integer. It leaves a remainder of 1 when divided by 2.

Step 1: Analyze statement (A) - "Even number \(\times\) even number = even number"
Let us multiply two general even integers, \(2n\) and \(2m\): \[ (2n) \times (2m) = 4nm = 2(2nm) \] Since the product possesses a clear factor of 2, the result is unconditionally an even number. *Example:* \(4 \times 6 = 24\) (which is Even). Thus, statement (A) is mathematically correct.

Step 2: Analyze statement (B) - "Even number \(\times\) odd number = even number"
Let us multiply a general even integer \(2n\) by a general odd integer \(2k + 1\): \[ (2n) \times (2k + 1) = 4nk + 2n = 2(2nk + n) \] Because the resulting expression can be completely factored by 2, the product remains even. *Example:* \(4 \times 5 = 20\) (which is Even). Thus, statement (B) is mathematically correct.

Step 3: Analyze statement (C) - "Odd number \(\times\) odd number = odd number"
Let us multiply two general odd integers, \((2n + 1)\) and \((2k + 1)\): \[ (2n + 1) \times (2k + 1) = 4nk + 2n + 2k + 1 = 2(2nk + n + k) + 1 \] The final form clearly matches the pattern \(2M + 1\), which defines an odd integer. *Example:* \(3 \times 5 = 15\) (which is Odd). Thus, statement (C) is mathematically correct.

Step 4: Analyze statement (D) - "Odd number \(\times\) even number = odd number"
Multiplication is commutative, meaning \(x \times y = y \times x\). Therefore, multiplying an odd number by an even number is identical to multiplying an even number by an odd number. As proven in Step 2, any real integer multiplication involving at least one even factor must result in an even product. \[ \text{Odd} \times \text{Even} = \text{Even} \] *Example:* \(3 \times 6 = 18\) (which is an even number, not an odd number). Consequently, statement (D) asserts an incorrect property, making it the right selection for this problem.