document.write( "Question 770086: if N is divided by 7 the remainder is 4 and if same number N is divided by 8 the remainder is 2. what is the value of N? \n" ); document.write( "

Algebra.Com's Answer #469312 by KMST(5328) $\"\"$ $\"About$

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From $\"7%2Am+%2B+2+=+8%2An\"$ you get
\n" ); document.write( " $\"7m%2B2=%287%2B1%29n%2B4\"$ --> $\"7m%2B2=7n%2Bn\"$ --> $\"7m-7n%2B2=n\"$ --> $\"n=7%28m-n%29%2B2\"$
\n" ); document.write( " $\"%28m-n%29%3C=-1\"$ would yield a negative $\"n\"$ ,
\n" ); document.write( " $\"n%3C=-5\"$ and a negative $\"N\"$ ,
\n" ); document.write( " $\"N=8n%2B4%3C=-20%2B4=-16\"$ and that does not make sense.
\n" ); document.write( " $\"%28m-n%29\"$ could be 0, 1, 2, 3, ...
\n" ); document.write( "For $\"m-n=0\"$ we get $\"n=7%2A0%2B2\"$ --> $\"n=2\"$ --> $\"N=8%2A2%2B2\"$ --> $\"highlight%28N=18%29\"$
\n" ); document.write( "But there is an infinite number of other solutions.
\n" ); document.write( " $\"m-n=1\"$ yields $\"n=7%2A1%2B2\"$ --> $\"n=9\"$ --> $\"N=8%2A9%2B2\"$ --> $\"highlight%28N=74%29\"$
\n" ); document.write( " $\"m-n=2\"$ yields $\"n=7%2A2%2B2\"$ --> $\"n=16\"$ --> $\"N=8%2A16%2B2\"$ --> $\"highlight%28N=130%29\"$
\n" ); document.write( "and so on.
\n" ); document.write( "After finding $\"n=18\"$ as one solution, we can keep getting more solutions $\"n\"$ by adding $\"56=7%2A8\"$ each time.
\n" ); document.write( "Since $\"56\"$ has a remainder of $\"0\"$ when divided by either $\"7\"$ or $\"8\"$ , adding $\"56\"$ or a multiple of $\"56\"$ does not change the remainders.
\n" ); document.write( "In general, we could rename $\"m-n\"$ as $\"p\"$ and say that $\"p\"$ is a non-negative integer.
\n" ); document.write( "Then $\"n=7p%2B2\"$ and $\"N=8%287p%2B2%29%2B2\"$ --> $\"N=56p%2B16%2B2\"$ --> $\"highlight%28N=56p%2B18%29\"$ \n" ); document.write( "

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