Proof That an Integer is Divisible by 2

[Math100]

Homework Statement

Establish the following divisibility criteria:
An integer is divisible by $2$ if and only if its units digit is $0, 2, 4, 6$, or $8$.

Relevant Equations

None.

Proof:

Suppose $N$ is the integer and $x$ is the units digit of $N$.
Then $N=10k+x$ for some $k\in\mathbb{Z}$ where $x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$.
Note that $10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2}$.
Thus $2\mid N\implies N\equiv 0\pmod {2}\implies x\equiv 0\pmod {2}\implies x\in{0, 2, 4, 6, 8}$.
Therefore, an integer is divisible by $2$ if and only if its units digit is $0, 2, 4, 6$, or $8$.

 

[fresh_42]

Homework Statement:: Establish the following divisibility criteria:
An integer is divisible by $2$ if and only if its units digit is $0, 2, 4, 6$, or $8$.
Relevant Equations:: None.

Proof:

Suppose $N$ is the integer and $x$ is the units digit of $N$.
Then $N=10k+x$ for some $k\in\mathbb{Z}$ where $x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$.
Note that $10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2}$.
Thus $2\mid N\implies N\equiv 0\pmod {2}\implies x\equiv 0\pmod {2}\implies x\in{0, 2, 4, 6, 8}$.
Therefore, an integer is divisible by $2$ if and only if its units digit is $0, 2, 4, 6$, or $8$.

Where is the other direction? You have shown $2\,|\,N \Longrightarrow x\in \{0,2,4,6,8\}$. You must also show that all integers that end on an even digit are even themselves. Of course, you simply could replace all $\Longrightarrow$ by $\Longleftrightarrow$.

 

[Math100]

Where is the other direction? You have shown $2\,|\,N \Longrightarrow x\in \{0,2,4,6,8\}$. You must also show that all integers that end on an even digit is even itself. Of course you simply could replace all $\Longrightarrow$ by $\Longleftrightarrow$.

So should I put "Thus $2\mid N\Leftrightarrow N\equiv 0\pmod {2}\Leftrightarrow x\equiv 0\pmod {2}\Leftrightarrow x\in{0, 2, 4, 6, 8}$.?

 

[fresh_42]

So should I put "Thus $2\mid N\Leftrightarrow N\equiv 0\pmod {2}\Leftrightarrow x\equiv 0\pmod {2}\Leftrightarrow x\in{0, 2, 4, 6, 8}$.?

Yes, the conclusions work in both directions because $10k \equiv 0 \pmod 2$ as you correctly said.