document.write( "Question 974052: find the greatest 4-digit number which when divided by 6,12,18,24 and 30 leaves5,11,17,23 and 29 as remainder respectively.
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Algebra.Com's Answer #596088 by Edwin McCravy(20060)![]() ![]() You can put this solution on YOUR website! find the greatest 4-digit number which when divided by 6,12,18,24 and 30 leaves 5,11,17,23 and 29 as remainder respectively. \n" ); document.write( " \r\n" ); document.write( "Theorem:\r\n" ); document.write( "\r\n" ); document.write( "Suppose k and n are positive integers.\r\n" ); document.write( "\r\n" ); document.write( "Then when kn-1 is divided by n the quotient is k-1 and the remainder is n-1.\r\n" ); document.write( "\r\n" ); document.write( "Proof:\r\n" ); document.write( " \r\n" ); document.write( " quotient\r\n" ); document.write( "divisor)dividend\r\n" ); document.write( " quotient×divisor\r\n" ); document.write( " remainder \r\n" ); document.write( "\r\n" ); document.write( "Or:\r\n" ); document.write( "\r\n" ); document.write( "dividend - (quotient)×(divisor) = remainder\r\n" ); document.write( " (kn-1) - (k-1)×(n) = n-1 \r\n" ); document.write( " \r\n" ); document.write( "\r\n" ); document.write( "Therefore 1 less than any common multiple of 6,12,18,24 and 30 will leave\r\n" ); document.write( "a remainder of 1 less than those, which are 5,11,17,23 and 29 respectively. \r\n" ); document.write( "\r\n" ); document.write( "The least common multiple of 6,12,18,24 and 30 is 360, so 1 less or 359 \r\n" ); document.write( "would be an answer if we didn't have the requirement that it be the greatest\r\n" ); document.write( "4-digit answer.\r\n" ); document.write( "\r\n" ); document.write( "However ANY multiple of 360 is also a multiple of 6,12,18,24 and 30. So\r\n" ); document.write( "we need only find the greatest 4-digit multiple of 360 and subtract 1.\r\n" ); document.write( "\r\n" ); document.write( "We divide the greatest 4 digit number 9999 by 360:\r\n" ); document.write( "\r\n" ); document.write( " 27\r\n" ); document.write( "360)9999\r\n" ); document.write( " 720\r\n" ); document.write( " 2799\r\n" ); document.write( " 2520\r\n" ); document.write( " 279\r\n" ); document.write( "\r\n" ); document.write( "So 27×360 = 9720 is the greatest 4-digit multiple of 6,12,18,24 and 30, and\r\n" ); document.write( "if we subtract 1, we get the greatest 4-digit number which leaves remainder\r\n" ); document.write( "5,11,17,23 and 29 respectively when divided by 6,12,18,24 and 30, respectively.\r\n" ); document.write( "\r\n" ); document.write( "Answer: 9720-1 = 9719\r\n" ); document.write( "\r\n" ); document.write( "Edwin\n" ); document.write( " |