Is there a proof or way of proving that all even numbers (taking into account the definition of an even number as [tex]n=2k[/tex]) end in 0,2,4,6, or 8?
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 [tex]n[/tex] is even if [tex]n=2k[/tex], for some integer [tex] k.[/tex] 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 ([tex]n=2k[/tex], where [tex]k[/tex] is an integer)?
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.