Show that the product of two even integers is even, the product of two odd integers is odd and the product of an even integer and an odd integer is even.

Step 1: Define two even integers Let's consider two even integers $a$ and $b$, where $a=2k$ and $b=2m$ for integers $k$ and $m$. Step 2: Simplify the product of two even integers Using the distributive property, you can simplify the product of the two even integers: $a\ast b=(2k)\ast (2m)$$a\ast b=4(km)$ Note that $4(km)$ can be written as $2(2km)$, where $2km$ is an integer since the product of integers is always an integer. This means the product of two even integers is also even. Step 3: Define two odd integers Now, let's take two odd integers $c$ and $d$, where $c=2n+1$ and $d=2p+1$ for integers $n$ and $p$. Step 4: Simplify the product of two odd integers Using the distributive property and combining like terms, you can simplify the product of the two odd integers: $c\ast d=(2n+1)\ast (2p+1)$$c\ast d=4np+2n+2p+1$ Since $4np$ and $2n$ and $2p$ are multiples of $2$, their sum $4np+2n+2p$ will also be a multiple of $2$. Therefore, we can rewrite the product as: $c\ast d=2(2np+n+p)+1$ Since $2np+n+p$ is an integer, the product of two odd integers is in the format $2q+1$ and thus, odd. Step 5: Define an even integer and an odd integer Lastly, let's consider an even integer $e$ and an odd integer $f$, where $e=2k$ and $f=2n+1$ for integers $k$ and $n$. Step 6: Simplify the product of an even integer and an odd integer Using the distributive property, you can simplify the product of the even and odd integers: $e\ast f=(2k)\ast (2n+1)$$e\ast f=4kn+2k$ Since $4kn$ and $2k$ are multiples of $2$, their sum $4kn+2k$ will also be a multiple of $2$. We can rewrite the product as: $e\ast f=2(2kn+k)$ Since $2kn+k$ is an integer, the product of an even integer and an odd integer is in the format $2r$ for $r$ as an integer, thus it is even. In conclusion, the product of two even integers is even, the product of two odd integers is odd, and the product of an even integer and an odd integer is even.