document.write( "Question 1045308: a school has fewer than 200 pupils
\n" ); document.write( "When they line up on rows of 4 there is 1 extra pupils\r
\n" ); document.write( "\n" ); document.write( "When they line up on rows of 5 there is 2 extra pupils\r
\n" ); document.write( "\n" ); document.write( "When they line up on rows of 6 there is 3 extra pupils\r
\n" ); document.write( "\n" ); document.write( "how many pupils could here be in the school?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #660710 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"N\"$ = number of pupils in the school.
\n" ); document.write( "When they line up on rows of 4 there is 1 extra pupil left over.
\n" ); document.write( "If $\"3\"$ more pupils were enrolled, there would be $\"N%2B3\"$ pupils,
\n" ); document.write( "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
\n" ); document.write( "would form another row of $\"3%2B1=4\"$ pupils.
\n" ); document.write( "So, let's enroll $\"3\"$ more pupils, because $\"N%2B3\"$ is a multiple of $\"4\"$ ,
\n" ); document.write( "and then we can line them all in rows of $\"4\"$ without leaving any extra pupils.
\n" ); document.write( "But what if we want to line them in rows of $\"5\"$ ?
\n" ); document.write( "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,
\n" ); document.write( "and they could form a new row of $\"3%2B2=5\"$ pupils.
\n" ); document.write( "It turns out that $\"N%2B3\"$ is a multiple of $\"5\"$ too.
\n" ); document.write( "All right, so $\"N%2B3\"$ is a good number of pupils if we want to line them up in rows of $\"4\"$ or $\"5\"$ ,
\n" ); document.write( "but what if we want to line them up in rows of $\"6\"$ ?
\n" ); document.write( "It did not work with $\"N\"$ pupils, because we had $\"3\"$ extra pupils left over.
\n" ); document.write( "With those $\"3\"$ pupils, and the $\"3\"$ newly enrolled pupils, we can make a new row with $\"3%2B3=6\"$ pupils.
\n" ); document.write( "Wow! It turns out that $\"N%2B3\"$ is a multiple of $\"6\"$ too.
\n" ); document.write( "So, $\"N%2B3\"$ is the right number of pupils for a school that likes to line them up in even rows.
\n" ); document.write( "That $\"N%2B3\"$ must be a magical number. It is a multiple of
\n" ); document.write( " $\"4\"$ , $\"5\"$ and $\"6\"$ .
\n" ); document.write( "The least common multiple of $\"4\"$ , $\"5\"$ and $\"6\"$ is
\n" ); document.write( " $\"60=6%2A10\"$ ,
\n" ); document.write( "the smaller multiple of $\"6\"$ that is also a multiple of $\"4\"$ , $\"5\"$ :
\n" ); document.write( " $\"60=4%2A15\"$ and $\"60=5%2A12\"$ .
\n" ); document.write( "So, it could be that
\n" ); document.write( " $\"N%2B3=60\"$ --> $\"N=60-3\"$ ---> $\"highlight%28N=57%29\"$ ,
\n" ); document.write( "or it could be that
\n" ); document.write( " $\"N%2B3=60%2A2=120\"$ --> $\"N=120-3\"$ ---> $\"highlight%28N=117%29\"$ ,
\n" ); document.write( "or it could be that
\n" ); document.write( " $\"N%2B3=60%2A3=180\"$ --> $\"N=180-3\"$ ---> $\"highlight%28N=177%29\"$ .
\n" ); document.write( "The next multiple of $\"60\"$ , $\"4%2A60=240\"$ , does not work,
\n" ); document.write( "because $\"N%2B3=240\"$ <---> $\"N=240-3=237\"$ makes the number of pupils greater than $\"200\"$ .
\n" ); document.write( "So the school could have $\"highlight%2857%29\"$ pupils, or $\"highlight%28117%29\"$ pupils, or $\"highlight%28177%29\"$ pupils. \n" ); document.write( "

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