document.write( "Question 404050: consider the following infinite sequence of numbers 17, 24, 31, 38, 45, 52. what is the least term in the sequence that is a multiple of 2, 3, 4, 5, 6? \n" ); document.write( "
Algebra.Com's Answer #285616 by Edwin McCravy(20059)\"\" \"About 
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document.write( "17, 24, 31, 38, 45, 52\r\n" );
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document.write( "That is an arithmetic sequence with a1 = 17 and d = 7\r\n" );
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document.write( "The nth term of an arithmetic sequence is given by\r\n" );
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document.write( "an = a1 + (n - 1)d\r\n" );
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document.write( "an = 17 + (n - 1)(7)\r\n" );
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document.write( "an = 17 + 7n - 7\r\n" );
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document.write( "an = 10 + 7n\r\n" );
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document.write( "To be a multiple of 2, 3, 4, 5, 6, is to be a multiple of 2*2*3*5 = 60\r\n" );
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document.write( "So we are looking for the smallest integer k so that\r\n" );
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document.write( "10 + 7n = 60k\r\n" );
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document.write( "Write the 10 and the 60 in terms of their nearest multiples of 7.\r\n" );
document.write( "10 is nearest 7 and 60 is nearest 63\r\n" );
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document.write( "(7 + 3) + 7n = (63 - 3)k\r\n" );
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document.write( "  7 + 3 + 7n = 63k - 3k\r\n" );
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document.write( "Divide through by 7\r\n" );
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document.write( "  1 + \"3%2F7\" + n = 9k - \"3%2F7\"k\r\n" );
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document.write( "Isolate the fractional terms:\r\n" );
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document.write( "\"3%2F7\" + \"3%2F7\"k = 9k - n - 1\r\n" );
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document.write( "Since the right side is an integer, the left side is also an integer.\r\n" );
document.write( "Let that integer be A.\r\n" );
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document.write( "\"3%2F7\" + \"3%2F7\"k = A    and   9k - n - 1 = A\r\n" );
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document.write( "Clear of fractions:\r\n" );
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document.write( "3 + 3k = 7A\r\n" );
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document.write( "Write the 7 in terms of its nearest multiple of 3, which is 6\r\n" );
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document.write( "3 + 3k = (6 + 1)A\r\n" );
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document.write( "3 + 3k = 6A + A\r\n" );
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document.write( "Divide through by 3\r\n" );
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document.write( "1 + k = 2A + \"A%2F3\"\r\n" );
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document.write( "Isolate the fraction:\r\n" );
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document.write( "1 + k - 2A = \"A%2F3\"\r\n" );
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document.write( "Since the left side is an integer, the right side is also an integer.\r\n" );
document.write( "Let that integer be B.\r\n" );
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document.write( "\"A%2F3\" = B     and   1 + k - 2A = B\r\n" );
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document.write( "Clear of fractions:\r\n" );
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document.write( "A = 3B\r\n" );
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document.write( "Substituting in 1 + k - 2A = B\r\n" );
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document.write( "1 + k - 2(3B) = B\r\n" );
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document.write( "1 + k = 6B + B\r\n" );
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document.write( "1 + k = 7B\r\n" );
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document.write( "    k = 7B - 1\r\n" );
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document.write( "Substituting for A and k in 9k - n - 1 = A\r\n" );
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document.write( "9(7B - 1) - n - 1 = 3B\r\n" );
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document.write( "  63B - 9 - n - 1 = 3B\r\n" );
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document.write( "     63B - 10 - n = 3B\r\n" );
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document.write( "         60B - 10 = n\r\n" );
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document.write( "The smallest possible value of B that would produce\r\n" );
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document.write( "a positive value for n is 1, thus B = 1\r\n" );
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document.write( "       60(1) - 10 = n\r\n" );
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document.write( "               50 = n\r\n" );
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document.write( "and k = 7B - 1 = 7(1) - 1 = 6\r\n" );
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document.write( "Therefore the desired term is the 50th.  That term can be\r\n" );
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document.write( "gotten from either \r\n" );
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document.write( "10 + 7n = 10 + 7(50) = 10 + 350 = 360\r\n" );
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document.write( "or from\r\n" );
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document.write( "60k = 60(6) = 360.\r\n" );
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document.write( "So the answer is 360, which is the 50th term.\r\n" );
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document.write( "Edwin
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