Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
Problem
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Tags: number theory
iniffur
16.05.2024 01:14
Sol.
$\frac{14n+25}{2n+1}=k^2\Longrightarrow 7+\frac{18}{2n+1}=k^2$
$\Longrightarrow 2n+1\in D_{18}=-18, -9, -6, -3,-2, -1,~1,~2,~3,~6,~9,~18 ~~$ (divisors of $18$)
$\Longrightarrow n\in \{-5,~-2,~-1,~0,~1,~4\}$
of which only $n=\boxed{-2,~0,~4}~$ make the given fraction a perfect square.
miyukina
16.05.2024 01:39
(14n + 25)/(2n + 1) = 7 + (25 – 1 × 7)/(2n + 1) = 7 + 18/(2n + 1) We need 18/(2n + 1) to equal { 2, 9, 18, –3, –6, –7 } , so 2n + 1 has to equal { 9, 2 , 1, –6, –3, –18/7 } Because 2n + 1 is an odd integer, then the solution is only for 2n + 1 = { 9, 1, –3 } with answers of n = { 4, 0, –2 }