document.write( "Question 1006513: find the number of common terms in two APs 7, 11, 15, .....up
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Algebra.Com's Answer #637981 by Edwin McCravy(20056) $\"\"$ $\"About$

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find the number of common terms in two APs 7, 11, 15, .....up
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document.write( "This may be a bit confusing because within each arithmetic\r\n" );
document.write( "sequence we will be dealing with another arithmetic sequence\r\n" );
document.write( "whose terms are the term numbers of the terms in each \r\n" );
document.write( "arithmetic sequence.\r\n" );
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document.write( "The Kth term of 7,11,15,... is 4K+3\r\n" );
document.write( "The Mth term of 9,16,23,... is 7M+2\r\n" );
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document.write( "When we set 4K+3 = 403 we get K=100 and when we set\r\n" );
document.write( "7M+2=702, we get M=100, so we are limiting both \r\n" );
document.write( "sequences to 100 terms. \r\n" );
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document.write( "We set their general terms equal\r\n" );
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document.write( " 4K+3 = 7M+2\r\n" );
document.write( "    1 = 7M-4K\r\n" );
document.write( "7M-4K = 1\r\n" );
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document.write( "The smaller coefficient in absolute value is 4, so we write\r\n" );
document.write( "the numbers in that Diophantine equation in terms of their\r\n" );
document.write( "nearest multiple of 4, including 0 as the nearest multiple \r\n" );
document.write( "of 4 to 1:\r\n" );
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document.write( "(8-1)M-4K = 1\r\n" );
document.write( "  8M-M-4K = 1\r\n" );
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document.write( "We divide every term by 4\r\n" );
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document.write( "2M-M/4-K = 1/4\r\n" );
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document.write( "Get fractions on the right side\r\n" );
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document.write( "2M-K = M/4+1/4\r\n" );
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document.write( "The left side is an integer and the right side is positive, so\r\n" );
document.write( "both sides are equal to a positive integer, say A.\r\n" );
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document.write( "2M-K=A;        M/4+1/4=A\r\n" );
document.write( "  -K=A-2M          M+1=4A\r\n" );
document.write( "   K=-A+2M           M=4A-1\r\n" );
document.write( "   K=2M-A            \r\n" );
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document.write( "Substitute 4A-1 for M in K=2M-A\r\n" );
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document.write( "   K=2(4A-1)-A\r\n" );
document.write( "   K=8A-2-A\r\n" );
document.write( "   K=7A-2\r\n" );
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document.write( "So for each positive integer A, term #(7A-2) of the first\r\n" );
document.write( "sequence equals term #(4A-1) of the second sequence.\r\n" );
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document.write( "7A-2 is an arithmetic sequence of term numbers of the first\r\n" );
document.write( "sequence.  It has common difference 7, and first term 5.\r\n" );
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document.write( "4A-1 is an arithmetic sequence of term numbers of the second\r\n" );
document.write( "sequence. It has common difference of 4, and first term 3.\r\n" );
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document.write( "That means the 5th term of the first sequence equals the 3rd\r\n" );
document.write( "term of the 2nd sequence.\r\n" );
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document.write( "And in general the arithmetic sequence of term numbers of the\r\n" );
document.write( "first sequence:\r\n" );
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document.write( "5th, 12th, 19th,...,(7A-2),...\r\n" );
document.write( "where A=1,2,3,...\r\n" );
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document.write( "equal respectively to the sequence of terms numbers of the \r\n" );
document.write( "second sequence:\r\n" );
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document.write( "3rd, 7th, 11th,...,(4A-1)...\r\n" );
document.write( "where A=1,2,3\r\n" );
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document.write( "Since we are limited by the 100th term of both sequences, and\r\n" );
document.write( "there are fewer term numbers (7A-2) in the first sequence than\r\n" );
document.write( "there are term numbers (4A-1) in the second sequence, we set \r\n" );
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document.write( "7A-2 <= 100\r\n" );
document.write( "  7A <= 102\r\n" );
document.write( "   A <= 14 4/7\r\n" );
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document.write( "Thus the number of term numbers of the terms of the first sequence \r\n" );
document.write( "which equal terms of the second sequence is the largest integer not\r\n" );
document.write( "exceeding 14 4/7 which is 14.\r\n" );
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document.write( "Answer: 14\r\n" );
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document.write( "Edwin

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