How is the product of two even numbers rewritten to show it is twice another quantity?

Conditions

• $x=2c$ and $y=2p$ with $c,p \in \mathbb{N}$.
• Basic arithmetic properties of multiplication (associativity and commutativity) are used.
• The goal is to express the product in the form $2k$ to show it is even.

Reasoning, step by step

1. Start with the product $xy$.
2. Substitute $x=2c$ and $y=2p$ to get $(2c)(2p)$.
3. Multiply the constants and collect the variables to get $4cp$.
4. Rewrite $4cp$ as $2(2cp)$ by factoring out a 2.
5. Conclude that the product is twice the quantity $2cp$, which has the algebraic form of an even number.

Example

The presenter writes `Product = xy`, then `= (2c)(2p)`, then `= 4cp`, and finally `= 2(2cp)`. The speaker says, "Now let's get the product. So the product of our two even numbers is xy, which is the same as 2c multiplied by 2p ... And 2c by 2p is 4cp ... another way to write 4 times cp would be twice 2cp."

Common misconceptions

• Thinking that $4cp$ is not even because it has a factor of 4.
• Forgetting to factor out the 2 to explicitly show the form $2k$.
• Believing the contradiction is fully derived within this specific clip (the clip ends before the explicit contradiction statement).

Watch the explanation

Explore next

• What exactly is assumed in the proof by contradiction for this theorem?
• How are the arbitrary even numbers represented algebraically in the proof?
• Why does the presenter switch from examples like $6\times 8$ and $2\times 4$ to an algebraic proof?