Question 1045308
{{{N}}} = number of pupils in the school.
When they line up on rows of 4 there is 1 extra pupil left over.
If {{{3}}} more pupils were enrolled, there would be {{{N+3}}} pupils,
and they could line up in rows of {{{4}}} without any extra pupil left over, because the {{{3}}} new pupils plus the {{{1}}} pupil previously left over
would form another row of {{{3+1=4}}} pupils.
So, let's enroll {{{3}}} more pupils, because {{{N+3}}} is a multiple of {{{4}}} ,
and then we can line them all in rows of {{{4}}} without leaving any extra pupils.
But what if we want to line them in rows of {{{5}}} ?
Well, that works too. We can group the {{{3}}} newly enrolled pupils with the {{{2}}} extra pupils that were left over when we only had {{{N}}} pupils,
and they could form a new row of {{{3+2=5}}} pupils.
It turns out that {{{N+3}}} is a multiple of {{{5}}} too.
All right, so {{{N+3}}} is a good number of pupils if we want to line them up in rows of {{{4}}} or {{{5}}} ,
but what if we want to line them up in rows of {{{6}}} ?
It did not work with {{{N}}} pupils, because we had {{{3}}} extra pupils left over.
With those {{{3}}} pupils, and the {{{3}}} newly enrolled pupils, we can make a new row with {{{3+3=6}}} pupils.
Wow! It turns out that {{{N+3}}} is a multiple of {{{6}}} too.
So, {{{N+3}}} is the right number of pupils for a school that likes to line them up in even rows.
That {{{N+3}}} must be a magical number. It is a multiple of
{{{4}}} , {{{5}}} and {{{6}}} .
The least common multiple of {{{4}}} , {{{5}}} and {{{6}}} is
{{{60=6*10}}} ,
the smaller multiple of {{{6}}} that is also a multiple of {{{4}}} , {{{5}}} :
{{{60=4*15}}} and {{{60=5*12}}} .
So, it could be that
{{{N+3=60}}} --> {{{N=60-3}}} ---> {{{highlight(N=57)}}} ,
or it could be that
{{{N+3=60*2=120}}} --> {{{N=120-3}}} ---> {{{highlight(N=117)}}} ,
or it could be that
{{{N+3=60*3=180}}} --> {{{N=180-3}}} ---> {{{highlight(N=177)}}} .
The next multiple of {{{60}}} , {{{4*60=240}}} , does not work,
because {{{N+3=240}}} <---> {{{N=240-3=237}}} makes the number of pupils greater than {{{200}}} .
So the school could have {{{highlight(57)}}} pupils, or {{{highlight(117)}}} pupils, or {{{highlight(177)}}} pupils.