Find-all-n-for-which-n-2-2n-4-is-divisible-by-7-

Question Number 133320 by bramlexs22 last updated on 21/Feb/21

Find all n for which n^{2} + 2 n + 4

is divisible by 7

Answered by EDWIN88 last updated on 21/Feb/21

let n = 7 k + r then n^{2} + 2 n + 4 = {(7 k + r)}^{2} + 2 (7 k + r) + 4

= r^{2} + 2 r + 4 (\mod 7)

cheking for r = 0, 1, 2, 3, 4, 5, 6

we see that r = 1 or r = 4

Hence n is of the form 7 k + 1 or 7 k + 4

i . e n \equiv 1 or 4 (\mod 7)

test n = 8 \Rightarrow n^{2} + 2 n + 4 = 64 + 16 + 4 = 84 is divisible by 7

test n = 11 \Rightarrow n^{2} + 2 n + 4 = 121 + 22 + 4 = 147 is divisible by 7

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