Question 1111064
According to my calculator,
500 is not divisible by 12.
500 divided by 12 is 41.6666.....
That means that {{{12*41}}} is
the largest multiple of 12 that is not more than 500,
or (in other words) 
{{{12*41}}} is the largest integer between 1 and 500 that is divisible by 12.
 
What about 200?
200 is not divisible by 12.
200 divided by 12 is 16.6666.....
That means that {{{12*16}}} is
the largest integer between 1 and 200 that is divisible by 12.
 
All integers between 1 and 500 are positive integers, of course,
but how many of them are between 200 and 500?
Well, there are {{{41}}} positive integers between 1 and 500:
{{{12*1}}} , {{{12*2}}} , {{{12*3}}} , ....... , {{{12*40}}} , and {{{12*41}}} .
However, the first {{{16}}} of those, from {{{12*1}}} to {{{12*16}}} , are less than 200,
so the number of integers between 200 and 500 are divisible by 12 is
{{{41-16=highlight(25)}}} .
How did we get that number?
We divided 500 by 12 and 200 by 12,
and we calculated the difference between the integer quotients
not considering remainders.
If using a calculator, we just used the integer part of the result.
 
The number of integers between 200 and 500 that are divisible by 6 is
{{{83-33=highlight(50)}}} ,
because 200 divided by 6 is 33.33333.... (or 33, remainder=2),
and 500 by 6 is 83.33333.... (or 83, remainder=2).
 
The case with integers divisible by 4 is trickier.
{{{200=4*50}}} is divisible by 4, and so is {{{500=4*125}}} .
Do they count as "between 200 and 500"?
If they count, we have to count the first {{{125}}} multiples of 4,
but subtract the first {{{49}}} to get {{{125-49=76}}} .
If neither 200 nor 500 is considered to be "between 200 and 500",
then there are {{{2}}} less, {{{76-2=74}}} numbers divisible by 4
between 200 and 500, not counting 200 or 500.
 
Similarly, the numbers between 200 and 500 that are divisible by {{{10}}}
are {{{50-19=31}}} if we count {{{200=10*20}}} and {{{500=10*50}}} , but they are {{{31-2=29}}} if we do not count either 200 or 500.