Question

how to solve : n square + n +1 is not divisible by 5

Answer 1

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