document.write( "Question 1209939: If m is a positive integer and sqrt(4m^2+29) is an integer, then what is m? \n" ); document.write( "
Algebra.Com's Answer #851125 by math_tutor2020(3816)\"\" \"About 
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\n" ); document.write( "Answer: m = 7\r
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\n" ); document.write( "\n" ); document.write( "Work Shown
\n" ); document.write( "k = some integer
\n" ); document.write( "k = sqrt(4m^2+29)
\n" ); document.write( "k^2 = 4m^2+29
\n" ); document.write( "k^2-4m^2 = 29
\n" ); document.write( "(k-2m)(k+2m) = 29
\n" ); document.write( "29 is prime so the only factors are 1 and 29.\r
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\n" ); document.write( "\n" ); document.write( "k and m are integers, so k-2m and k+2m are also integers.
\n" ); document.write( "In order for (k-2m)(k+2m) = 29 to be the case, we'd need either
\n" ); document.write( "\"system%28k-2m+=+1%2Ck%2B2m=29%29\"
\n" ); document.write( "or
\n" ); document.write( "\"system%28k-2m+=+29%2Ck%2B2m=1%29\"
\n" ); document.write( "In either system of equations, adding straight down leads to 2k = 30 which solves to k = 15.\r
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\n" ); document.write( "\n" ); document.write( "Then,
\n" ); document.write( "k-2m = 1
\n" ); document.write( "15-2m = 1
\n" ); document.write( "-2m = 1-15
\n" ); document.write( "-2m = -14
\n" ); document.write( "m = -14/(-2)
\n" ); document.write( "m = 7
\n" ); document.write( "Or,
\n" ); document.write( "k-2m = 29
\n" ); document.write( "15-2m = 29
\n" ); document.write( "-2m = 29-15
\n" ); document.write( "-2m = 14
\n" ); document.write( "m = 14/(-2)
\n" ); document.write( "m = -7
\n" ); document.write( "We ignore m = -7 since your teacher stated that m > 0
\n" ); document.write( "Check: sqrt(4m^2+29)=sqrt(4*7^2+29)=15 is an integer.
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