document.write( "Question 1209235: Find the number of ordered pairs (m,n) of integers that satisfy
\n" ); document.write( "mn = 3m + 3n + 17. \n" ); document.write( "

Algebra.Com's Answer #847883 by AnlytcPhil(1806) $\"\"$ $\"About$

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document.write( "Long division:\r\n" );
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document.write( "        3\r\n" );
document.write( "n-3)3n+17\r\n" );
document.write( "    3n- 9\r\n" );
document.write( "       26\r\n" );
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document.write( "n-3 must divide evenly into 26, so n-3 is either ±1,±2,±13, or ±26\r\n" );
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document.write( "That's 8 possibilities, so we know the answer is 8. To check, we find them:\r\n" );
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document.write( "1. if n-3 = -26, then n = -23, and m = 3+26/(-23-3) = 2, ordered pair (2,-23).\r\n" );
document.write( "2. if n-3 = -13, then n = -10, and m = 3+26/(-10-3) = 1, ordered pair (1,-10).\r\n" );
document.write( "3. if n-3 = -2, then n = 1, and m = 3+26/(1-3) = -10, ordered pair (-10,1).\r\n" );
document.write( "4. if n-3 = -1, then n = 2, and m = 3+26/(2-3) = -23, ordered pair (-23,2).\r\n" );
document.write( "5. if n-3 = 1, then n = 4, and m = 3+26/(4-3) = 29, ordered pair (29,4).\r\n" );
document.write( "6. if n-3 = 2, then n = 5, and m = 3+26/(5-3) = 16, ordered pair (16,5).\r\n" );
document.write( "7. if n-3 = 13, then n = 16, and m = 3+26/(16-3) = 5, ordered pair (5,16).\r\n" );
document.write( "8. if n-3 = 26, then n = 29, and m = 3+26/(29-3) = 4, ordered pair (4,29).\r\n" );
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document.write( "So indeed, there are 8 such ordered pairs.\r\n" );
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document.write( "Here is a plot of all 8.  See how nicely they fit on that curve.\r\n" );
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document.write( "Can you guess what the equation of that curve is?\r\n" );
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document.write( "Edwin

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