Math Challenges

Submissions for Problem #12

Problem #12

Prove that the product of two even numbers is an even number.

Prove that the product of two odd numbers is an odd number.

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fish-face

Solution: \( \displaylines{2n\cdot2m=\left(2\cdot2\right)\left(n\cdot m\right)\left(\text{commutativity}\right)\\ =2\cdot\left(2\cdot n\cdot m\right)\left(\text{associativity}\right)\\ \left(2n+1\right)\left(2m+1\right)=\left(2n+1\right)2m+\left(2n+1\right)1\left(\text{distributivity}\right)\\ =2n2m+2m+2n+1\left(\text{distributivity,identity}\right)\\ =2\left(2nm+m+n\right)+1\left(\text{distributivity}\right)} \)

zapwai

Solution: \( \displaylines{\left(2r\right)\left(2s\right)=4rs\text{ which is even.}\\ \left(2m+1\right)\left(2n+1\right)=4mn+2\left(m+n\right)+1=2\left(2mn+m+n\right)+1\text{ (odd)}} \)

Explanation:
Another way to think about it is with factors - an even number multiplied by anything will still have 2 as a factor, and is thus even. Odd numbers do not contain 2 as a factor, and neither would their product.