Find-the-maximum-possible-integer-n-such-that-n-1-n-2-n-3-n-2-4-is-also-an-integer-

Question Number 110593 by Aina Samuel Temidayo last updated on 29/Aug/20

Find the maximum possible integer n

such that \frac{(n - 1) (n^{2} + n - 3)}{n^{2} + 4} is also an

integer

Answered by floor(10²Eta[1]) last updated on 19/Oct/20

n^{2} + 4 ∣ (n - 1) (n^{2} + n - 3) = n^{3} - 4 n + 3

\Leftrightarrow n^{2} + 4 ∣ n (n^{2} + 4) - (n^{3} - 4 n + 3) = 8 n - 3

∴ 8 n \equiv 3 (\mod n^{2} + 4)

\Rightarrow {64 n}^{2} \equiv 9 (\mod n^{2} + 4)

n^{2} \equiv - 4 (\mod n^{2} + 4)

\Rightarrow {64 n}^{2} \equiv - 256 (\mod n^{2} + 4)

∴ 9 \equiv - 256 (\mod n^{2} + 4)

\Rightarrow n^{2} + 4 ∣ 265

n^{2} + 4 \in {5, 53}

n^{2} + 4 = 5 \Rightarrow n = \pm 1

n^{2} + 4 = 53 \Rightarrow n = 7

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

I see . {You}^{'} ve corrected your mistake .

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

Thanks .