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\n" ); document.write( "The equation obviously has an infinite number of solutions; choose any values you want for two of the variables and determine the corresponding value of the third.
\n" ); document.write( "So for the problem to make sense, we need to assume we are looking for non-negative integer solutions, or perhaps for positive integer solutions. Allowing a value of 0 for any of the variables produces a large number of additional solutions, so we will assume we are looking for solutions in positive integers.
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\n" ); document.write( "2x, 2z, and 16 are all even; that means 3y must be even; and that means y most be even. Since the sum is 16, that means the only possible values for y are 2 and 4.
\n" ); document.write( "If y=2, then
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\n" ); document.write( "In positive integers, that gives us 4 solutions:
\n" ); document.write( "(x,y,z)=(1,2,4)
\n" ); document.write( "(x,y,z)=(2,2,3)
\n" ); document.write( "(x,y,z)=(3,2,2)
\n" ); document.write( "(x,y,z)=(4,2,1)
\n" ); document.write( "If y=4, then
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\n" ); document.write( "In positive integer, that gives us only one solution:
\n" ); document.write( "(x,y,z)=(1,4,1)
\n" ); document.write( "So we have found a total of five solutions to the equation in positive integers.
\n" ); document.write( "Since there are no instructions in the statement of the problem that let us choose one of the five solutions, all five of them should be considered acceptable.
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