![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
\r\n" );
document.write( "That is wrong for 890 does not have three consecutive digits.\r\n" );
document.write( "He should have ruled that case out.\r\n" );
document.write( "\r\n" );
document.write( "There are two possible cases for the 3 digit number n, with three\r\n" );
document.write( "consecutive digits.\r\n" );
document.write( "\r\n" );
document.write( "Case 1:\r\n" );
document.write( "\r\n" );
document.write( "hundreds digit = h\r\n" );
document.write( "tens digit = h+1\r\n" );
document.write( "hundreds digit = h+2\r\n" );
document.write( "\r\n" );
document.write( "100h + 10(h+1) + h+2 = 100h + 10h + 10 + h+2 = 111h + 12 \r\n" );
document.write( "\r\n" );
document.write( "This must be equal to the sum of 4 multiples of 10.\r\n" );
document.write( "The sum of 4 multiples of 10 is itself a multiple of 10.\r\n" );
document.write( "Suppose A is such that this multiple of 10 is 10A. Then\r\n" );
document.write( "\r\n" );
document.write( "111h + 12 = 10A\r\n" );
document.write( "\r\n" );
document.write( "Write each number in terms of its nearest multiple of 10\r\n" );
document.write( "\r\n" );
document.write( "(110+1)h + 10+2 = 10A\r\n" );
document.write( "\r\n" );
document.write( "110h + h + 10 + 2 = 10A\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 10\r\n" );
document.write( "\r\n" );
document.write( "11h + h/10 + 1 + 2/10 = A\r\n" );
document.write( "\r\n" );
document.write( "h/10 + 2/10 = A - 11h - 2 \r\n" );
document.write( "\r\n" );
document.write( "The right side is an integer so the left side must\r\n" );
document.write( "also be an integer, say B.\r\n" );
document.write( "\r\n" );
document.write( "h/10 + 2/10 = B \r\n" );
document.write( "\r\n" );
document.write( "h + 2 = 10B\r\n" );
document.write( " h = 10B - 2\r\n" );
document.write( "\r\n" );
document.write( "Since h is a digit, 0 through 9, then B can only be 1\r\n" );
document.write( " h = 10(1)-2 = 10-2 = 8\r\n" );
document.write( "\r\n" );
document.write( "But if h=8, then the tens digit is 9 and the hundreds\r\n" );
document.write( "digit would be 10, but 10 is not a digit. So case 1\r\n" );
document.write( "is eliminated.\r\n" );
document.write( "\r\n" );
document.write( "The other tutor should have eliminated this case.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So we can only consider case 2.\r\n" );
document.write( "\r\n" );
document.write( "Case 2:\r\n" );
document.write( "\r\n" );
document.write( "ones digit = u\r\n" );
document.write( "tens digit = u+1\r\n" );
document.write( "hundreds digit = u+2\r\n" );
document.write( "\r\n" );
document.write( "100(u+2) + 10(u+1) + u = 100u + 200 + 10u + 10 + u = 111u + 210 \r\n" );
document.write( "\r\n" );
document.write( "This must be equal to the sum of 4 multiples of 10.\r\n" );
document.write( "The sum of 4 multiples of 10 is itself a multiple of 10.\r\n" );
document.write( "Suppose A is such that this multiple of 10 is 10A. \r\n" );
document.write( "Then the sum of the 4 multiples of 10 is 10A\r\n" );
document.write( "\r\n" );
document.write( "111u + 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( "Write 111 in terms of its nearest multiple of 10, which is 110.\r\n" );
document.write( "\r\n" );
document.write( "(110+1)u + 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( "110u + u + 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 10\r\n" );
document.write( "\r\n" );
document.write( "11u + u/10 + 21 = A\r\n" );
document.write( "\r\n" );
document.write( "u/10 = A - 11u - 21\r\n" );
document.write( "\r\n" );
document.write( "Since the right side is an integer, so is the\r\n" );
document.write( "left side. Suppose that integer is B, then\r\n" );
document.write( "\r\n" );
document.write( "u/10 = B\r\n" );
document.write( "\r\n" );
document.write( "u = 10B\r\n" );
document.write( "\r\n" );
document.write( "But u is a digit, and the only digit that is\r\n" );
document.write( "a multiple of 10 is 0, so u = 0\r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "ones digit = u = 0\r\n" );
document.write( "tens digit = u+1 = 1\r\n" );
document.write( "hundreds digit = u+2 = 2\r\n" );
document.write( "\r\n" );
document.write( "So the 3 digit number is 210. <-- the only interesting answer!\r\n" );
document.write( " Too bad you weren't asked that. \r\n" );
document.write( "\r\n" );
document.write( "But sadly, you were asked what are the 4 multiples of 10.\r\n" );
document.write( "\r\n" );
document.write( "[Really! They should have asked you what is the 3-digit number. \r\n" );
document.write( "That's the interesting answer! For the 3 digit number must \r\n" );
document.write( "be 210 regardless of what the four multiples of 10 are]\r\n" );
document.write( "\r\n" );
document.write( "But let's find what you were asked for, uninteresting as it is.\r\n" );
document.write( "\r\n" );
document.write( "The sum of those 4 multiples of 10 is 10A,\r\n" );
document.write( "so we substitute u=0\r\n" );
document.write( "\r\n" );
document.write( " 111u + 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( "111(0) + 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( " 210 = 10A\r\n" );
document.write( "\r\n" );
document.write( "So any 4 multiples of 10 that have sum 210 will\r\n" );
document.write( "be an answer to the question.\r\n" );
document.write( "\r\n" );
document.write( "The answer is any of these 120 groups of 4 multiples \r\n" );
document.write( "of 10 that have sum 210. They are all listed below.\r\n" );
document.write( "Be sure to tell your teacher that it would have \r\n" );
document.write( "been a million times more interesting to have \r\n" );
document.write( "asked the question \"What is the 3 digit number?\", \r\n" );
document.write( "the answer which would have been 210 regardless \r\n" );
document.write( "of which of the 120 groups of 4 multiples of 10 \r\n" );
document.write( "could have been chosen. But here are all 120\r\n" );
document.write( "answers:\r\n" );
document.write( "\r\n" );
document.write( "1. {0,0,0,210}\r\n" );
document.write( "2. {0,0,10,200}\r\n" );
document.write( "3. {0,0,20,190}\r\n" );
document.write( "4. {0,0,30,180}\r\n" );
document.write( "5. {0,0,40,170}\r\n" );
document.write( "6. {0,0,50,160}\r\n" );
document.write( "7. {0,0,60,150}\r\n" );
document.write( "8. {0,0,70,140}\r\n" );
document.write( "9. {0,0,80,130}\r\n" );
document.write( "10. {0,0,90,120}\r\n" );
document.write( "11. {0,0,100,110}\r\n" );
document.write( "12. {0,10,10,190}\r\n" );
document.write( "13. {0,10,20,180}\r\n" );
document.write( "14. {0,10,30,170}\r\n" );
document.write( "15. {0,10,40,160}\r\n" );
document.write( "16. {0,10,50,150}\r\n" );
document.write( "17. {0,10,60,140}\r\n" );
document.write( "18. {0,10,70,130}\r\n" );
document.write( "19. {0,10,80,120}\r\n" );
document.write( "20. {0,10,90,110}\r\n" );
document.write( "21. {0,10,100,100}\r\n" );
document.write( "22. {0,20,20,170}\r\n" );
document.write( "23. {0,20,30,160}\r\n" );
document.write( "24. {0,20,40,150}\r\n" );
document.write( "25. {0,20,50,140}\r\n" );
document.write( "26. {0,20,60,130}\r\n" );
document.write( "27. {0,20,70,120}\r\n" );
document.write( "28. {0,20,80,110}\r\n" );
document.write( "29. {0,20,90,100}\r\n" );
document.write( "30. {0,30,30,150}\r\n" );
document.write( "31. {0,30,40,140}\r\n" );
document.write( "32. {0,30,50,130}\r\n" );
document.write( "33. {0,30,60,120}\r\n" );
document.write( "34. {0,30,70,110}\r\n" );
document.write( "35. {0,30,80,100}\r\n" );
document.write( "36. {0,30,90,90}\r\n" );
document.write( "37. {0,40,40,130}\r\n" );
document.write( "38. {0,40,50,120}\r\n" );
document.write( "39. {0,40,60,110}\r\n" );
document.write( "40. {0,40,70,100}\r\n" );
document.write( "41. {0,40,80,90}\r\n" );
document.write( "42. {0,50,50,110}\r\n" );
document.write( "43. {0,50,60,100}\r\n" );
document.write( "44. {0,50,70,90}\r\n" );
document.write( "45. {0,50,80,80}\r\n" );
document.write( "46. {0,60,60,90}\r\n" );
document.write( "47. {0,60,70,80}\r\n" );
document.write( "48. {0,70,70,70}\r\n" );
document.write( "49. {10,10,10,180}\r\n" );
document.write( "50. {10,10,20,170}\r\n" );
document.write( "51. {10,10,30,160}\r\n" );
document.write( "52. {10,10,40,150}\r\n" );
document.write( "53. {10,10,50,140}\r\n" );
document.write( "54. {10,10,60,130}\r\n" );
document.write( "55. {10,10,70,120}\r\n" );
document.write( "56. {10,10,80,110}\r\n" );
document.write( "57. {10,10,90,100}\r\n" );
document.write( "58. {10,20,20,160}\r\n" );
document.write( "59. {10,20,30,150}\r\n" );
document.write( "60. {10,20,40,140}\r\n" );
document.write( "61. {10,20,50,130}\r\n" );
document.write( "62. {10,20,60,120}\r\n" );
document.write( "63. {10,20,70,110}\r\n" );
document.write( "64. {10,20,80,100}\r\n" );
document.write( "65. {10,20,90,90}\r\n" );
document.write( "66. {10,30,30,140}\r\n" );
document.write( "67. {10,30,40,130}\r\n" );
document.write( "68. {10,30,50,120}\r\n" );
document.write( "69. {10,30,60,110}\r\n" );
document.write( "70. {10,30,70,100}\r\n" );
document.write( "71. {10,30,80,90}\r\n" );
document.write( "72. {10,40,40,120}\r\n" );
document.write( "73. {10,40,50,110}\r\n" );
document.write( "74. {10,40,60,100}\r\n" );
document.write( "75. {10,40,70,90}\r\n" );
document.write( "76. {10,40,80,80}\r\n" );
document.write( "77. {10,50,50,100}\r\n" );
document.write( "78. {10,50,60,90}\r\n" );
document.write( "79. {10,50,70,80}\r\n" );
document.write( "80. {10,60,60,80}\r\n" );
document.write( "81. {10,60,70,70}\r\n" );
document.write( "82. {20,20,20,150}\r\n" );
document.write( "83. {20,20,30,140}\r\n" );
document.write( "84. {20,20,40,130}\r\n" );
document.write( "85. {20,20,50,120}\r\n" );
document.write( "86. {20,20,60,110}\r\n" );
document.write( "87. {20,20,70,100}\r\n" );
document.write( "88. {20,20,80,90}\r\n" );
document.write( "89. {20,30,30,130}\r\n" );
document.write( "90. {20,30,40,120}\r\n" );
document.write( "91. {20,30,50,110}\r\n" );
document.write( "92. {20,30,60,100}\r\n" );
document.write( "93. {20,30,70,90}\r\n" );
document.write( "94. {20,30,80,80}\r\n" );
document.write( "95. {20,40,40,110}\r\n" );
document.write( "96. {20,40,50,100}\r\n" );
document.write( "97. {20,40,60,90}\r\n" );
document.write( "98. {20,40,70,80}\r\n" );
document.write( "99. {20,50,50,90}\r\n" );
document.write( "100. {20,50,60,80}\r\n" );
document.write( "101. {20,50,70,70}\r\n" );
document.write( "102. {20,60,60,70}\r\n" );
document.write( "103. {30,30,30,120}\r\n" );
document.write( "104. {30,30,40,110}\r\n" );
document.write( "105. {30,30,50,100}\r\n" );
document.write( "106. {30,30,60,90}\r\n" );
document.write( "107. {30,30,70,80}\r\n" );
document.write( "108. {30,40,40,100}\r\n" );
document.write( "109. {30,40,50,90}\r\n" );
document.write( "110. {30,40,60,80}\r\n" );
document.write( "111. {30,40,70,70}\r\n" );
document.write( "112. {30,50,50,80}\r\n" );
document.write( "113. {30,50,60,70}\r\n" );
document.write( "114. {30,60,60,60}\r\n" );
document.write( "115. {40,40,40,90}\r\n" );
document.write( "116. {40,40,50,80}\r\n" );
document.write( "117. {40,40,60,70}\r\n" );
document.write( "118. {40,50,50,70}\r\n" );
document.write( "119. {40,50,60,60}\r\n" );
document.write( "120. {50,50,50,60}\r\n" );
document.write( "\r\n" );
document.write( "Edwin \n" );
document.write( "