Question 1045308: a school has fewer than 200 pupils
When they line up on rows of 4 there is 1 extra pupils
When they line up on rows of 5 there is 2 extra pupils
When they line up on rows of 6 there is 3 extra pupils
how many pupils could here be in the school?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  = number of pupils in the school.
When they line up on rows of 4 there is 1 extra pupil left over.
If  more pupils were enrolled, there would be  pupils,
and they could line up in rows of  without any extra pupil left over, because the new pupils plus the  pupil previously left over
would form another row of  pupils.
So, let's enroll more pupils, because is a multiple of ,
and then we can line them all in rows of without leaving any extra pupils.
But what if we want to line them in rows of  ?
Well, that works too. We can group the newly enrolled pupils with the  extra pupils that were left over when we only had pupils,
and they could form a new row of  pupils.
It turns out that is a multiple of too.
All right, so is a good number of pupils if we want to line them up in rows of or ,
but what if we want to line them up in rows of  ?
It did not work with pupils, because we had extra pupils left over.
With those pupils, and the newly enrolled pupils, we can make a new row with  pupils.
Wow! It turns out that is a multiple of too.
So, is the right number of pupils for a school that likes to line them up in even rows.
That must be a magical number. It is a multiple of
, and .
The least common multiple of , and is
 ,
the smaller multiple of that is also a multiple of , :
 and  .
So, it could be that
 -->  --->  ,
or it could be that
 -->  --->  ,
or it could be that
 -->  --->  .
The next multiple of  ,  , does not work,
because  <--->  makes the number of pupils greater than  .
So the school could have  pupils, or  pupils, or  pupils.
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