Show that n^2+n+1 is not divisible by 5 for any n where n is a natural number
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A
Show that n 2+n+1 is not divisible by 5 for any n where n is a natural number
Ask for details Follow Report by Msharma7865 01.02.2018
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cutedoll01
cutedoll01 Genius
Any natural number it's "
n = 5k
n = 5k±1
n = 5k±2
where k natural number k
Case n = 5k then
n^2+n+1 = 25k^2+5k+1 = 5k(k+1)+1 then not divisible by 5 (remainder 1)
CAse n = 5k±1 then
n^2+n+1 = 25k^2 ± 10k+1 + 5k±1+1 = 5k(5k±1) +2±1 then not divisible by 5 (remainder 3 or 1)
CAse n = 5k±2
n^2+n+1 = 25k^2 ± 10k+4 + 5k±2+1 = 5k(5k±1+1) ± 2 then not divisible by 5 (remainder 2 or 3)
Then
n^2+n+1 not divisible by 5
B
If n^2+n+1 divisible by 5 then
exist k like
n^2+n+1=5k then
n^2+n+1-5k=0
n zeroes (root) of n^2+n+1-5k=0
If n natural first n must be real then integer then positive
Discriminant =
D= sqrt(1-4+20k)=sqrt(20k-3)
20k-3 must be perfect square let be m^2 then
20k-3 = m^2
k = (1/20)(m^2+3)
m^2 +3 must dividing by 10 then m^2 must finishing in ...7
No perfect square finishing in 7 then no m no k then n it's irational then no exist natural n