Prove algebraically that the product of an even number and an odd number is always even.

We have defined even and odd numbers mathematically and demonstrated that the product of an even number and an odd number results in an even number by showing the product can be factored into the form 2 n , where n is an integer. Thus, the statement is proven algebraically. Therefore, the product of an even number and an odd number is always even.
;

Answered by Anonymous | 2025-07-04

Represent an even number as 2 k and an odd number as 2 m + 1 , where k and m are integers.
Multiply the even and odd numbers: ( 2 k ) ( 2 m + 1 ) = 4 km + 2 k .
Factor out a 2: 4 km + 2 k = 2 ( 2 km + k ) .
Since 2 ( 2 km + k ) is in the form 2 n where n is an integer, the product is always even: The product is always even. ​

Explanation

Defining Even and Odd Numbers Let's start by defining what even and odd numbers are in mathematical terms. An even number can be expressed as 2 k , where k is any integer. Similarly, an odd number can be expressed as 2 m + 1 , where m is any integer. Our goal is to prove that when we multiply an even number by an odd number, the result is always an even number.

Multiplying Even and Odd Numbers Now, let's multiply the even number 2 k by the odd number 2 m + 1 :
( 2 k ) ( 2 m + 1 ) We will use the distributive property to expand this expression.

Expanding the Expression Expanding the expression, we get: 4 km + 2 k Now, we want to show that this expression is always even. To do this, we can factor out a 2 from the expression:

Factoring and Defining a New Integer Factoring out a 2, we have: 2 ( 2 km + k ) Let's define a new integer n such that n = 2 km + k . Then, our expression becomes: 2 n Since n is an integer, 2 n is, by definition, an even number.

Conclusion Therefore, the product of an even number and an odd number is always even. We have shown that ( 2 k ) ( 2 m + 1 ) = 2 n , where n is an integer. This completes the proof.


Examples
Understanding that the product of an even and an odd number is even is useful in various scenarios. For instance, if you're arranging seating at an event and need an even number of seats per table, knowing that an even number of tables with any number of odd-numbered groups will still result in an even total number of attendees simplifies planning. Similarly, in computer science, this principle is used in parity checks for error detection, where ensuring the number of '1' bits is even helps verify data integrity.

Answered by GinnyAnswer | 2025-07-04