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14. Show that n2 + n+ 1 is not divisible by 5 for any n, where n is a natural number. |
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Answer» 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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