Why numbers with 0 as it's unit digit are even?

Question

I wanted to know why if the number ends with

0, 2, 4, 6, or 8 is even and if starts with 1,3, 5, 7, or 9 is odd

I think if we know why if the number endswith 0 is always even will answer this

because from addition rules

even + even = even even + odd = odd

so always we have to split the number into sum for ex:

5425 = 5420 + 5

which is even + odd = odd

but why having 5420 0 at the beginning made it even number

++++++++++++++++++++++ edit:

why this rule doesn't apply on numbers which are divisible by 3 or 4

Answer 1

It's fairly simple.

If a numbers is the the digits $abcd$ (I'm arbitrarily using $4$ digits to make this east to think about; the same argument will apply for any number of digits.

And $abcd = 1000a + 100b +10c + a$ or in other words $abcd = 10abc + d$.

If you divide by $2$ You get $(abcd)\div 2 = 5abc + (d\div 2)$. $5abc$ is a whole number because $2$ divides into $10$ evenly. And so $2$ will divide evenly into $abcd$ if and only if $2$ divides into $d$ with no remainder.

This will only work for numbers that divide into $10$ evenly.

If you take a number that doesn't divide evenly into $10$ it won't work.

$(abcd)\div 3 = \frac{10}3abc + (d\div 3)$ but $3$ *doesn't divide $\frac{10}3$ evenly so we don't know if $10abc\div 3$ will have a remainder or not.

There are other rules we can do but they are not so simple.

...... example;

$abcd \div 3 = (1000a + 100b + 10c +d)\div 3 = (999a + a + 99b + b + 9c + c + d)\div 3=$

$(333a + 33b + 3c) + (a+b+c+d)\div 3$.

So $abcd$ is divisible by $3$ if and only if $a+b+c+d$ is divisible by $3$.

But that rule only works for numbers that divide $9$ (that is $3$ and $9$).

We can make other rules that are more complicated.

$abcd= 100ab + 10c + d$ so

$abcd\div 4 = 25ab + (10c + d)\div 4$.

If $c$ is even then $10c\div 4 = 5\frac c2$ has no divisor and $4$ divides $abcd$ if $c$ is even and $4$ divides $d$.

If $c$ is odd, then $10c = 8c + 2c$ divided by $4$ will have remainder $2$ and $4$ divides $abcd$ if $c$ is odd and $4$ divides $d + 2$. Or in other words if $d$ is even but $4$ does not divide $d$.

So $4$ divides $abcd$ if $c$ is even and $d$ is divisble by $4$ or if $c$ is odd and $d$ is even but not divisible by $4$.

That's a rule that works with $4$ but not with other numbers in general.

Answer 2

Thanks to all comments

the answer as i got is :

for the first section:-

all the numbers that has unit digit 0 are divisibe by 10 which is divisible by 2 so it's even

for the second section:- i think it's about the Cyclicity of the numbers 3 and 4

if the number is divisible by 3 by division theorem

x = 3*r for any r in N

if we traced the Cyclicity of 3 we will find that

3*0 = 0 , 3*1 = 3 , 3*2 = 6 , 3*3 = 9 .....etc

it will be 0,3,6,9,2,5,8,1,4,7

which are not always even or add

if someone have any comment please tell :)