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\n" ); document.write( "King David has two sets of numbers.
\n" ); document.write( "The first sets starts with 7 and follows a pattern of increasing by 6.
\n" ); document.write( "The second sets starts with 5554 and follows a pattern decreasing by 7.
\n" ); document.write( "How many numbers do the two sets have in common ?
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document.write( "First set is {a(n)} = { 7+6n | n= 0,1,2,3, . . . }. (1)\r\n" );
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document.write( "Second set is {b(m)} = { 5554-7m | m= 0,1,2,3, . . . }. (2)\r\n" );
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document.write( "For convenience, I wil present the second sequence in INCREASING order, too, by analogy with the first sequence.\r\n" );
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document.write( "Then the second sequence is {b(m) = { 3+7m | m= 0,1,2,3, . . . }.\r\n" );
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document.write( " (I chose a \"3\" as the starting point for the second sequence.\r\n" );
document.write( " The value of \"3\" is the minimum positive number of the sequence (2),\r\n" );
document.write( " and I start the sequece (2) from \"3\", since I am going to look for common terms with the sequense (1), \r\n" );
document.write( " which starts from 7).\r\n" );
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document.write( "To start, I prepared Excel spreadsheet, which is shown below.\r\n" );
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document.write( " Sequence 2 Sequense 1\r\n" );
document.write( "m b(m) n a(n)\r\n" );
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document.write( "0 3 0 7 \r\n" );
document.write( "1 10 1 13 \r\n" );
document.write( "2 17 2 19 \r\n" );
document.write( "3 24 3 25 \r\n" );
document.write( "4 31 4 31 (*)\r\n" );
document.write( "5 38 5 37 \r\n" );
document.write( "6 45 6 43 \r\n" );
document.write( "7 52 7 49 \r\n" );
document.write( "8 59 8 55 \r\n" );
document.write( "9 66 9 61 \r\n" );
document.write( "10 73 10 67 (**) \r\n" );
document.write( "11 80 11 73 (**) \r\n" );
document.write( "12 87 12 79 \r\n" );
document.write( "13 94 13 85 \r\n" );
document.write( "14 101 14 91 \r\n" );
document.write( "15 108 15 97 \r\n" );
document.write( "16 115 16 103 \r\n" );
document.write( "17 122 17 109 \r\n" );
document.write( "18 129 18 115 \r\n" );
document.write( "19 136 19 121 \r\n" );
document.write( "20 143 20 127 \r\n" );
document.write( "21 150 21 133 \r\n" );
document.write( "22 157 22 139 \r\n" );
document.write( "23 164 23 145 \r\n" );
document.write( "24 171 24 151 \r\n" );
document.write( "25 178 25 157 \r\n" );
document.write( "26 185 26 163 \r\n" );
document.write( "27 192 27 169 \r\n" );
document.write( "28 199 28 175 \r\n" );
document.write( "29 206 29 181 \r\n" );
document.write( "30 213 30 187 \r\n" );
document.write( "31 220 31 193 \r\n" );
document.write( "32 227 32 199 \r\n" );
document.write( "33 234 33 205 \r\n" );
document.write( "34 241 34 211 \r\n" );
document.write( "35 248 35 217 \r\n" );
document.write( "36 255 36 223 \r\n" );
document.write( "37 262 37 229 \r\n" );
document.write( "38 269 38 235 \r\n" );
document.write( "39 276 39 241 \r\n" );
document.write( "40 283 40 247 \r\n" );
document.write( "41 290 41 253 \r\n" );
document.write( "42 297 42 259 \r\n" );
document.write( "43 304 43 265 \r\n" );
document.write( "44 311 44 271 \r\n" );
document.write( "45 318 45 277 \r\n" );
document.write( "46 325 46 283 \r\n" );
document.write( "47 332 47 289 \r\n" );
document.write( "48 339 48 295 \r\n" );
document.write( "49 346 49 301 \r\n" );
document.write( "50 353 50 307 \r\n" );
document.write( "51 360 51 313 \r\n" );
document.write( "52 367 52 319 \r\n" );
document.write( "53 374 53 325 \r\n" );
document.write( "54 381 54 331 \r\n" );
document.write( "55 388 55 337 \r\n" );
document.write( "56 395 56 343 \r\n" );
document.write( "57 402 57 349 \r\n" );
document.write( "58 409 58 355 \r\n" );
document.write( "59 416 59 361 \r\n" );
document.write( "60 423 60 367 \r\n" );
document.write( "61 430 61 373 \r\n" );
document.write( "62 437 62 379 \r\n" );
document.write( "63 444 63 385 \r\n" );
document.write( "64 451 64 391 \r\n" );
document.write( "65 458 65 397 \r\n" );
document.write( "66 465 66 403 \r\n" );
document.write( "67 472 67 409 \r\n" );
document.write( "68 479 68 415 \r\n" );
document.write( "69 486 69 421 \r\n" );
document.write( "70 493 70 427 \r\n" );
document.write( "71 500 71 433 \r\n" );
document.write( "72 507 72 439 \r\n" );
document.write( "73 514 73 445 \r\n" );
document.write( "74 521 74 451 \r\n" );
document.write( "75 528 75 457 \r\n" );
document.write( "76 535 76 463 \r\n" );
document.write( "77 542 77 469 \r\n" );
document.write( "78 549 78 475 \r\n" );
document.write( "79 556 79 481 \r\n" );
document.write( "80 563 80 487 \r\n" );
document.write( "81 570 81 493 \r\n" );
document.write( "82 577 82 499 \r\n" );
document.write( "83 584 83 505 \r\n" );
document.write( "84 591 84 511 \r\n" );
document.write( "85 598 85 517 \r\n" );
document.write( "86 605 86 523 \r\n" );
document.write( "87 612 87 529 \r\n" );
document.write( "88 619 88 535 \r\n" );
document.write( "89 626 89 541 \r\n" );
document.write( "90 633 90 547 \r\n" );
document.write( "91 640 91 553 \r\n" );
document.write( "92 647 92 559 \r\n" );
document.write( "93 654 93 565 \r\n" );
document.write( "94 661 94 571 \r\n" );
document.write( "95 668 95 577 \r\n" );
document.write( "96 675 96 583 \r\n" );
document.write( "97 682 97 589 \r\n" );
document.write( "98 689 98 595 \r\n" );
document.write( "99 696 99 601 \r\n" );
document.write( "100 703 100 607 \r\n" );
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document.write( "By inspection, the first common term is the number 31: a( 4) = 7+6*4 = 31, b( 4) = 3 + 7*4 = 31 (marked by (*) in the Table).\r\n" );
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document.write( " The second common term is the number 73: a(11) = 7+6*11 = 73, b(10) = 3 + 7*10 = 73 (marked by (**) in the Table).\r\n" );
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document.write( "Notice that 73 - 31 = 42 = 6*7.\r\n" );
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document.write( "From now, it is clear that the common terms form an arithmetic progression\r\n" );
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document.write( "with the first term of 31 and the common difference of 42.\r\n" );
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document.write( "To find the number of such common terms, we divide the interval [31,5554] of the length (5554-30) = 5524\r\n" );
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document.write( "by equal intervals of the length 42: 
= 131.52.\r\n" );
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document.write( " +-------------------------------------------------+\r\n" );
document.write( " | So, we have 131 full interval, which means |\r\n" );
document.write( " | that the number of common terms is 132. |\r\n" );
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document.write( "Thus the number of the common terms of sequences (1) and (2) is 132.\r\n" );
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document.write( "The greatest common term is 31 + 42*(132-1) = 5533. \r\n" );
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document.write( "ANSWER. There are 132 common terms. They form an arithmetic progression.\r\n" );
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document.write( " First common term is 31; the common difference is 42. The last common term is 31 + 42*131 = 5533.\r\n" );
document.write( "\r\n" ); document.write( "\n" ); document.write( "Solved.\r
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