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Olympiad workbook
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IOQM
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Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

A positive integer m has the property that m^2 is expressible in...

A positive integer mm has the property that m2m^2 is expressible in the form 4n2−5n+164 n^2-5 n+16 where nn is an integer (of any sign). Find the maximum possible value of ∣m−n∣|m-n|.

Correct answer

14

Step-by-step solution

∵m2=4n2−5n+16∴4n2−5n+16−m2=0∴8n=5±16m2−231\begin{array}{ll} \because & m^2=4 n^2-5 n+16 \\ \therefore & 4 n^2-5 n+16-m^2=0 \\ \therefore & 8 n=5 \pm \sqrt{16 m^2-231} \end{array} Let D2=16m2−231D^2=16 m^2-231 and D>0D>0 ⇒(4m+D)(4m−D)=231=7×11×13\Rightarrow (4 m+D)(4 m-D)=231=7 \times 11 \times 13 Case-I: 4m+D=231 and 4m−D=14 m+D=231 \text { and } 4 m-D=1 then m=29m=29 and D=115,n=15D=115, n=15 ∴∣m−n∣=14\therefore \quad|m-n|=14 Case-II: 4m+D=77 and 4m−D=3∴m=10,D=37 and n=−4∴∣m−n∣=14\begin{array}{ll} & 4 m+D=77 \text { and } 4 m-D=3 \\ \therefore & m=10, D=37 \text { and } n=-4 \\ \therefore & |m-n|=14 \end{array} Case-III: 4m+D=33 and 4m−D=7∴m=5,D=13 and n=−1∴∣m−n∣=6\begin{array}{ll} & 4 m+D=33 \text { and } 4 m-D=7 \\ \therefore & m=5, D=13 \text { and } n=-1 \\ \therefore & |m-n|=6 \end{array} Case-IV: 4m+D=21 and 4m−D=11∴m=4,D=5 and n=0\begin{array}{ll} & 4 m+D=21 \text { and } 4 m-D=11 \\ \therefore & m=4, D=5 \text { and } n=0 \end{array} Hence ∣m−n∣=4|m-n|=4 Maximum value of ∣m−n∣=14|m-n|=14
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