document.write( "Question 768603: Find the greatest number of 6 digit dividing by 15,18,21,24,27 leaves the remainder 10,13,16,19,and 22. \n" ); document.write( "

Algebra.Com's Answer #468434 by KMST(5328) $\"\"$ $\"About$

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The remainders are always 5 short of the divisor.
\n" ); document.write( "For example, if the number, $\"N\"$ is divided by 15, the remainder is $\"10\"$ , so that for some integer $\"p\"$ ,
\n" ); document.write( " $\"N=15p%2B10\"$ --> $\"N%2B5=15p%2B15=15%28p%2B1%29\"$ ,
\n" ); document.write( "which makes $\"N%2B5\"$ a multiple of $\"15\"$ .
\n" ); document.write( "The same happens with 18, 21, 24, and 27.
\n" ); document.write( "That number, $\"N\"$ , is such that $\"N%2B5\"$ is a multiple of 15, 18, 21, 24, and 27.
\n" ); document.write( "That means $\"N%2B5\"$ is a multiple of the least common multiple of
\n" ); document.write( " $\"15=3%2A5\"$ ,
\n" ); document.write( " $\"18=3%2A2%2A3=2%2A3%5E2\"$ ,
\n" ); document.write( " $\"21=3%2A7\"$ ,
\n" ); document.write( " $\"24=3%2A2%5E3\"$ , and
\n" ); document.write( " $\"27=3%5E3\"$ ,
\n" ); document.write( "and that least common multiple is
\n" ); document.write( " $\"2%5E3%2A3%5E3%2A5%2A7=7560\"$
\n" ); document.write( "Dividing 1,000,000 by 7560 we find a quotient of 132 plus a remainder of 2080.
\n" ); document.write( "That means that $\"132%2A7560%2B2080=1000000\"$ --> $\"132%2A7560=1000000-2080=997920\"$
\n" ); document.write( "and 997,920 is a multiple of 7560.
\n" ); document.write( "The next multiple of 7560 is $\"133%2A7560=1005480\"$
\n" ); document.write( "That means that $\"997920\"$ is the largest $\"N%2B5\"$ multiple of 15, 18, 21, 24, and 27 that will yield a number $\"N\"$ with 6 digits,
\n" ); document.write( "So $\"N%2B5=997920\"$ --> $\"highlight%28N=997915%29\"$ .
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