document.write( "Question 847901: Find the least number which when divided by 6, 8, and 15 leaves a remainder 5, but when divided by 13, leaves no remainder. \n" ); document.write( "
Algebra.Com's Answer #510748 by Edwin McCravy(20064)\"\" \"About 
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document.write( "Find the least number which when divided by 6, 8, and 15 leaves a remainder 5, but when divided by 13, leaves no remainder.\r\n" );
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document.write( "Let the number be N\r\n" );
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document.write( "N is 5 more than a multiple of 6, so N=6p+5 for some integer p.\r\n" );
document.write( "N is 5 more than a multiple of 8, so N=8q+5 for some integer q.\r\n" );
document.write( "N is 5 more than a multiple of 15, so N=15r+5 for some integer r.\r\n" );
document.write( "N is a multiple of 13, so N=13t\r\n" );
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document.write( "So\r\n" );
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document.write( "(1)   N = 6p+5 = 8q+5 = 15r+5 = 13t\r\n" );
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document.write( "Since N = 15r+5 = 5(3r+1), is a multiple of 13, 3r+1 is also multiple of 13.\r\n" );
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document.write( "So 3r+1 = 13u for some integer u\r\n" );
document.write( "     3r = 13u-1\r\n" );
document.write( "So  15r = 65u-5\r\n" );
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document.write( "So (1) becomes\r\n" );
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document.write( "N = 6p+5 = 8q+5 = 65u-5+5 = 13t\r\n" );
document.write( "N = 6p+5 = 8q+5 = 65u = 13t\r\n" );
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document.write( "We can dispense with the 13t, since N is a multiple of 65, it's \r\n" );
document.write( "automatically a multiple of 13. So (1) becomes\r\n" );
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document.write( "(2)   N = 6p+5 = 8q+5 = 65u\r\n" );
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document.write( "Since 6p+5 = 8q+5\r\n" );
document.write( "        6p = 8q\r\n" );
document.write( "        3p = 4q\r\n" );
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document.write( "Write 4q as 3q+q\r\n" );
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document.write( "        3p = 3q+q\r\n" );
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document.write( "Divide through by 3\r\n" );
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document.write( "         p = q+\"q%2F3\"\r\n" );
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document.write( "So     p-q = \"q%2F3\"\r\n" );
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document.write( "Since p-q is an integer, so is \"q%2F3\", say the integer v\r\n" );
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document.write( "Then p-q = v and \"q%2F3\" = v\r\n" );
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document.write( "So \"q%2F3\" = v\r\n" );
document.write( "   q = 3v\r\n" );
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document.write( "p-q = v\r\n" );
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document.write( "p-3v = v\r\n" );
document.write( "   p = 4v \r\n" );
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document.write( "Substituting in (2)\r\n" );
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document.write( "   N = 6(4v)+5 = 8(3v)+5 = 65u\r\n" );
document.write( "   N = 24v+5 = 65u\r\n" );
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document.write( "Since  24v+5 = 65u\r\n" );
document.write( "         24v = 65u-5\r\n" );
document.write( "         24v = 5(13u-1)\r\n" );
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document.write( "v is a multiple of 5, so v = 5w for some integer w\r\n" );
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document.write( "      24(5w) = 5(13u-1)\r\n" );
document.write( "         14w = 13u-1\r\n" );
document.write( "           1 = 13u-14w   \r\n" );
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document.write( "Write 14 as 13+1\r\n" );
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document.write( "           1 = 13u-(13+1)w\r\n" );
document.write( "           1 = 13u-13w-w\r\n" );
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document.write( "Divide through by 13\r\n" );
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document.write( "          \"1%2F13\" = u-w-\"w%2F13\"\r\n" );
document.write( "     \"1%2F13%2Bw%2F13\" = u-w\r\n" );
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document.write( "The right side is an integer so the left side must be also.\r\n" );
document.write( "Suppose that integer is x.  Then\r\n" );
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document.write( "\"1%2F13%2Bw%2F13\" = x and u-w = x\r\n" );
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document.write( "1 + w = 13x\r\n" );
document.write( "    w = 13x-1\r\n" );
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document.write( "Substituting in \r\n" );
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document.write( "  u-w = x\r\n" );
document.write( "  u-(13x-1) = x\r\n" );
document.write( "  u-13x+1 = x\r\n" );
document.write( "       u = 14x-1 \r\n" );
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document.write( "Since N = 65u\r\n" );
document.write( "      N = 65(14x-1)\r\n" );
document.write( "      N = 910x-65\r\n" );
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document.write( "The smallest value x can take on is 1, so\r\n" );
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document.write( "      N = 910(1)-65\r\n" );
document.write( "      N = 910-65\r\n" );
document.write( "      N = 845 \r\n" );
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document.write( "Edwin
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