document.write( "Question 1074625: Find all positive integers $n$ for which $n^2 - 19n + 99$ is a perfect square. \n" ); document.write( "

Algebra.Com's Answer #689544 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
TIP:
\n" ); document.write( "When I see $ wrapping equations,
\n" ); document.write( "it remind me of LATEX,
\n" ); document.write( "and makes me suspect you come from artofproblemsolving.
\n" ); document.write( "We use triple curly brackets, like {, here.
\n" ); document.write( "I am going to look at the embarrassingly simpler solution in that website
\n" ); document.write( "as soon as I finish typing my awkward solution here.
\n" ); document.write( "
\n" ); document.write( "ONE WAY:
\n" ); document.write( "We want $\"n%5E2-19n%2B99=K%5E2\"$ to be true
\n" ); document.write( "for some positive integers $\"n\"$ and $\"k\"$ .
\n" ); document.write( " $\"n%5E2-19n%2B99=K%5E2\"$
\n" ); document.write( " $\"n%5E2-19n-K%5E2=-99\"$
\n" ); document.write( " $\"n%5E2-19n%2B19%5E2%2F4-K%5E2=19%5E2%2F4-99\"$
\n" ); document.write( " $\"%28n-19%2F2%29%5E2-K%5E2=19%5E2%2F4-99\"$
\n" ); document.write( " $\"4%28n-19%2F2%29%5E2-4K%5E2=19%5E2-4%2A99\"$
\n" ); document.write( " $\"%282n-19%29%5E2-%282K%29%5E2=361-396\"$
\n" ); document.write( " $\"%282n-19%29%5E2-%282K%29%5E2=-35\"$
\n" ); document.write( " $\"%282K%29%5E2-%282n-19%29%5E2=35\"$
\n" ); document.write( "Let me abbreviate with the change of variables
\n" ); document.write( " $\"system%28a=2K%2Cb=2n-19%29\"$ ,
\n" ); document.write( "knowing that $\"a=2K%3E0\"$ is a positive, even integer,
\n" ); document.write( "and $\"b\"$ is an integer.
\n" ); document.write( "The equation becomes
\n" ); document.write( " $\"a%5E2-b%5E2=35\"$ , so much easier to write,
\n" ); document.write( "and immediately we think of
\n" ); document.write( " $\"%28a%2Bb%29%28a-b%29=35\"$ .
\n" ); document.write( "Are those factors in brackets positive or negative?
\n" ); document.write( "We knew that $\"a%5E2-b%5E2=35%3E0\"$ ,
\n" ); document.write( "so $\"a%5E2%3Eb%5E2\"$ --> $\"sqrt%28a%5E2%29%3Esqrt%28b%5E2%29\"$ --> $\"a%3Eabs%28b%29\"$ ,
\n" ); document.write( "so whether $\"b%3E0\"$ or $\"b%3C0\"$ ,
\n" ); document.write( " $\"%28a%2Bb%29%3E0\"$ and $\"%28a%2Bb%29%3E0\"$ .
\n" ); document.write( "Back to the product of positive integer factors
\n" ); document.write( " $\"%28a%2Bb%29%28a-b%29=35\"$ .
\n" ); document.write( "The factors could be 1 and 35 or 5 and 7.
\n" ); document.write( "We have 4 possibilities:
\n" ); document.write( " $\"system%28a%2Bb=1%2Ca-b=35%29\"$ --> $\"system%28a=18%2Cb=-17%29\"$ --> $\"2n-19=-17\"$ --> $\"highlight%28n=1%29\"$
\n" ); document.write( " $\"system%28a%2Bb=35%2Ca-b=1%29\"$ --> $\"system%28a=18%2Cb=17%29\"$ --> $\"2n-19=17\"$ --> $\"highlight%28n=18%29\"$
\n" ); document.write( " $\"system%28a%2Bb=5%2Ca-b=7%29\"$ --> $\"system%28a=6%2Cb=-2%29\"$ --> $\"2n-19=-1\"$ --> $\"highlight%28n=9%29\"$
\n" ); document.write( " $\"system%28a%2Bb=7%2Ca-b=5%29\"$ --> $\"system%28a=6%2Cb=1%29\"$ --> $\"2n-19=1\"$ --> $\"highlight%28n=10%29\"$
\n" ); document.write( "
\n" ); document.write( "ANOTHER WAY:
\n" ); document.write( " $\"y=x%5E2-19x%2B99\"$ <--> $\"y=%28x-9.5%29%5E2%2B8.75\"$ , so if $\"y=K%5E2\"$ ,
\n" ); document.write( " $\"K%5E2=%28x-9.5%29%5E2%2B8.75\"$ --> $\"K%5E2-%28x-9.5%29%5E2=8.75\"$ --> $\"%282K%29%5E2-%282x-19%29%5E2=4%2A8.75=35\"$ .
\n" ); document.write( "The two perfect squares we look for, $\"%282K%29%5E2\"$ and $\"%282x-19%29%5E2\"$ ,
\n" ); document.write( "cannot be too large if their difference is only $\"35\"$ .
\n" ); document.write( "There are not many possible solutions,
\n" ); document.write( "and they will all be close to the $\"x=9.5\"$ of the vertex.
\n" ); document.write( "So, listing values for $\"y=n%5E2-19n%2B99\"$ around $\"n=9.5\"$ ,
\n" ); document.write( "(or listing values for $\"%282n-19%29%5E2%2B35\"$ for $\"2n\"$ around $\"n=9.5\"$ ),
\n" ); document.write( "until you find a perfect square is an option.
\n" ); document.write( "
\n" ); document.write( "PS: Actually, it was not that the artofproblemsolving solution was so much simpler,
\n" ); document.write( "but that they skip steps/explanations that \"should be obvious,\"
\n" ); document.write( "so they do not need to write as much.
\n" ); document.write( "If you get all those steps, artofproblemsolving is a fun site,
\n" ); document.write( "but I tend to/like to overexplain, so I belong at this website. \n" ); document.write( "

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