Something interesting concerning parity

[sEsposito]

Is there a proof or way of proving that all even numbers (taking into account the definition of an even number as $$n=2k$$) end in 0,2,4,6, or 8?

 

[Gokul43201]

What do you think? Care to try showing that numbers ending in 1,3,5,7, or 9 can not be even?

 

[TwilightTulip]

Rewrite n as 10a+b, where 0<=b<10.

Then n is even means 2 divides 10a+b, i.e. 2 divides b (let 10a+b=2k and solve for b). Hence b=0,2,4,6,8

 

[sEsposito]

Rewrite n as 10a+b, where 0<=b<10.

Then n is even means 2 divides 10a+b, i.e. 2 divides b (let 10a+b=2k and solve for b). Hence b=0,2,4,6,8

An explanation such as this is alright for proving that an integer $$n$$ is even if $$n=2k$$, for some integer $$k.$$ Thus, we have to use our knowledge of even numbers to say that they end in an integer that is divisible by 2. A satisfactory explanation, no doubt, just not what I'm looking for.

Is there a way to show that the fact that all even numbers end in 0,2,4,6, or 8 implies the definition of an even number ($$n=2k$$, where $$k$$ is an integer)?

 

[Prologue]

Any number that is a multiple of ten is even and divisible by 2. Then any number (base ten) denoted by ...dcba (where a is the ones place, b in the tens, etc.) will be divisible by two if a is divisible by 2 because ...+d*10^3+c*10^2+b*10^1+a represents the number and division is linear. All terms except a are multiples of ten always and therefore divisible by 2, then all that is left is a.