Why doesn't the last digit method work for divisibility by 4? I wish to explain to a grade 5 kid, the answer to the question - 'When can we know that the observing the last digit of a number will not be sufficient to determine its divisibility by a given N?'
For example - It doesn't work for 3, because we get all sorts of digits, i.e. from 0 to 9, at the units place. The same reason works for 7 and 9. 
But what would be a convincing argument for 4 and 8? Even though the last digits form a pattern - 0, 2, 4, 6 and 8 - but we're still not sure about its divisibility.
 A: Any positive integer $N$ can be written as $$100a+10b+c$$
$$=4(25a+2b)+2b+c$$
So, we need $2b+c$ to be divisible by $4\iff N$ is to be divisible by $4$
Another observation: Clearly $c$  must be even $=2d$(say)
So, we need $b+d$ to be even i.e., $b,d$ must of same parity i.e., either both even or both odd
A: I think we have use example to explain this to a Grade 5 kid.
$$2316=23\times 100+16=23\times (25\times 4)+16=(23\times 25)\times 4+16$$
$16$ is divisible by $4$ because $16=4\times 4$. This also means that
$$2316=(23\times 25)\times 4+4\times 4=(23\times 25+4)\times 4$$
and it is divisible by $4$.
On the other hand,
$$2317=(23\times 25)\times 4+17$$
$17$ is not divisible by $4$ since $17=4\times 4+1$ (we have a non-zero remainder). This also means that
$$2317=(23\times 25)\times 4+4\times 4+1=(23\times 25+4)\times 4+1$$
and it is not divisible by $4$, as we have a non-zero remainder.
A: To understand how the divisibility test for $4$ works it's helpful to think about how we write numbers, specifically in terms of a base $10$ expansion.  For example, when we write something like $1234$ what we really mean is $$1000 + 200 + 30 + 4.$$  Everything past the $10$'s place is automatically divisible by $4$, because $100$, $1000$, etc. are all divisible by $4.$  The $10$'s place and the $1$'s place are where we need to check, that is we need to check if the last two digits are divisible by $4$ because that is the part of the number that is not automatically divisible by $4$ when thinking about it in terms of  a base $10$ expansion.  In this example $1234$ is not divisible by $4$ because while both $1000$ and $200$ are divisible by $4$, $34$ is not.  On the other hand $1236$ would be divisible by $4$ because $36$ is.
