document.write( "Question 1157079: Find the ordered pair (m,n), where m,n are positive integers satisfying the following equation:
\n" ); document.write( "\"14mn=55-7m-2n\" \n" ); document.write( "
Algebra.Com's Answer #779897 by greenestamps(13203)\"\" \"About 
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\n" ); document.write( "Here is a solution using the method of Diophantine equations.

\n" ); document.write( "A basic Diophantine equation is a single equation with two unknowns, in which the solutions are restricted to integers. The requirement that the solutions be integers limits the solutions to only a few, or, as in this case, to a single solution.

\n" ); document.write( "This example is a very common type, containing terms in m, n, and mn. The basic techniques for solving that kind of equation are standard.

\n" ); document.write( "(1) Solve the equation for one of the variables in terms of the other.

\n" ); document.write( "\"14mn+=+55-7m-2n\"
\n" ); document.write( "\"14mn%2B7m+=+55-2n\"
\n" ); document.write( "\"7m%282n%2B1%29+=+55-2n\"
\n" ); document.write( "\"7m+=+%2855-2n%29%2F%282n%2B1%29\"

\n" ); document.write( "(2) Find the value(s) of n that make the fraction an integer.

\n" ); document.write( "m has to be a positive integer, so 7m is an integer. That means \"%2855-2n%29%2F%282n%2B1%29\" is an integer.

\n" ); document.write( "It is generally, as in this case, difficult to find values of n that make that expression an integer. So this is what we usually do at this point: write the numerator of the fraction as a multiple of the denominator plus a constant. It's as if we are doing the division and expressing the fraction as an integer plus a remainder.

\n" ); document.write( "\"7m+=+%2856-1-2n%29%2F%282n%2B1%29\"
\n" ); document.write( "\"7m+=+%2856-%281%2B2n%29%29%2F%282n%2B1%29\"
\n" ); document.write( "\"7m+=+56%2F%282n%2B1%29-1\"

\n" ); document.write( "Now 7m and 1 are integers, so \"56%2F%282n%2B1%29\"must be an integer; and that means 2n+1 must be a divisor of 56.

\n" ); document.write( "In general, we would look at all the factors of 56 and see what values of n would make \"56%2F%282n%2B1%29\" an integer.

\n" ); document.write( "But in this example we can go directly to the solution with a bit of logical reasoning.

\n" ); document.write( "For all n, 2n+1 is odd; so 2n+1 must be an odd divisor of 56.

\n" ); document.write( "But there is only one odd divisor of 56: 7. (Okay, 1 is an odd divisor of 56; but that would make n=0, and the problem requires m and n to be positive integers.)

\n" ); document.write( "So 2n+1 = 7, which makes n=3.

\n" ); document.write( "Then \"7m+=+56%2F7-1+=+8-1+=+7\" --> m = 1.

\n" ); document.write( "ANSWER: (m,n) = (1,3)

\n" ); document.write( "CHECK:
\n" ); document.write( "14mn = 14*1*3 = 42
\n" ); document.write( "55-7m-2n = 55-7-6 = 42
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