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 #847882 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( " $\"mn=3m%2B3n%2B17\"$

\n" ); document.write( "Solve the equation for one variable in terms of the other:

\n" ); document.write( " $\"mn-3m=3n%2B17\"$
\n" ); document.write( " $\"m%28n-3%29=3n%2B17\"$
\n" ); document.write( " $\"m=%283n%2B17%29%2F%28n-3%29\"$

\n" ); document.write( "Perform the indicated division and express the result as quotient and remainder:

\n" ); document.write( " $\"m=%28%283n-9%29%2B26%29%2F%28n-3%29\"$
\n" ); document.write( " $\"m=3%2B26%2F%28n-3%29\"$

\n" ); document.write( "In that last equation, m and 3 are integers, so $\"26%2F%28n-3%29\"$ must be an integer.

\n" ); document.write( "The number of ordered pair solutions is the number of integer factors of 26, which is 4.

\n" ); document.write( "NOTE: In typical problems like this, we are looking for solutions in positive integers. However, this problem does not specify positive integers; counting positive and negative integers, the number of integer factors of 26 is 8.

\n" ); document.write( "So there are 8 ordered pair solutions.

\n" ); document.write( "ANSWER: 8

\n" ); document.write( "The problem doesn't ask us to find the solutions, but we can do so to verify that there are 4 pairs of solutions. Note that the expression is symmetrical in m and n, so if (a,b) is a solution the (b,a) will be a solution. So to find the 8 solutions we only need to find 4 of them and switch the order of the two numbers to get the other solutions.

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document.write( "   n-3  n  m=3+26/(n-3)   solutions (m,n)\r\n" );
document.write( "  ---------------------------------------\r\n" );
document.write( "    1   4   3+26/1 = 29     (4,29) and (29,4)\r\n" );
document.write( "    2   5   3+26/2 = 16     (5,16) and (16,5)\r\n" );
document.write( "   -1   2   3+26/-1 = -23   (2,-23) and (-23,2)\r\n" );
document.write( "   -2   1   3+26/-2 = -10   (1,-10) and (-10,1)

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