Question

plz solve this question ( only question number 2 )

Answer 1

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_________"METHOD ----> A"_________



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 


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_______"METHOD ----------> 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

Answer 2

question is totally unclear and cannot be read.