TITLE: Is $x^2+x+1$ divisible by $101$, if $x\in\mathbb Z$?
QUESTION [5 upvotes]: Prove $x^2+x+1$ isn't divisible by $101$, for any $x\in\mathbb Z$?
I think the way of solving the problem it by using "Fermat's Little Theorem".

REPLY [4 votes]: I'm probably gonna get down-voted for this, but here is one way to prove it:

$x\equiv  0\pmod{101} \implies x^2+x+1\equiv  0^2+  0+1\equiv  1\pmod{101}$
$x\equiv  1\pmod{101} \implies x^2+x+1\equiv  1^2+  1+1\equiv  3\pmod{101}$
$x\equiv  2\pmod{101} \implies x^2+x+1\equiv  2^2+  2+1\equiv  7\pmod{101}$
$x\equiv  3\pmod{101} \implies x^2+x+1\equiv  3^2+  3+1\equiv 13\pmod{101}$
$x\equiv  4\pmod{101} \implies x^2+x+1\equiv  4^2+  4+1\equiv 21\pmod{101}$
$x\equiv  5\pmod{101} \implies x^2+x+1\equiv  5^2+  5+1\equiv 31\pmod{101}$
$x\equiv  6\pmod{101} \implies x^2+x+1\equiv  6^2+  6+1\equiv 43\pmod{101}$
$x\equiv  7\pmod{101} \implies x^2+x+1\equiv  7^2+  7+1\equiv 57\pmod{101}$
$x\equiv  8\pmod{101} \implies x^2+x+1\equiv  8^2+  8+1\equiv 73\pmod{101}$
$x\equiv  9\pmod{101} \implies x^2+x+1\equiv  9^2+  9+1\equiv 91\pmod{101}$
$x\equiv 10\pmod{101} \implies x^2+x+1\equiv 10^2+ 10+1\equiv 10\pmod{101}$
$x\equiv 11\pmod{101} \implies x^2+x+1\equiv 11^2+ 11+1\equiv 32\pmod{101}$
$x\equiv 12\pmod{101} \implies x^2+x+1\equiv 12^2+ 12+1\equiv 56\pmod{101}$
$x\equiv 13\pmod{101} \implies x^2+x+1\equiv 13^2+ 13+1\equiv 82\pmod{101}$
$x\equiv 14\pmod{101} \implies x^2+x+1\equiv 14^2+ 14+1\equiv  9\pmod{101}$
$x\equiv 15\pmod{101} \implies x^2+x+1\equiv 15^2+ 15+1\equiv 39\pmod{101}$
$x\equiv 16\pmod{101} \implies x^2+x+1\equiv 16^2+ 16+1\equiv 71\pmod{101}$
$x\equiv 17\pmod{101} \implies x^2+x+1\equiv 17^2+ 17+1\equiv  4\pmod{101}$
$x\equiv 18\pmod{101} \implies x^2+x+1\equiv 18^2+ 18+1\equiv 40\pmod{101}$
$x\equiv 19\pmod{101} \implies x^2+x+1\equiv 19^2+ 19+1\equiv 78\pmod{101}$
$x\equiv 20\pmod{101} \implies x^2+x+1\equiv 20^2+ 20+1\equiv 17\pmod{101}$
$x\equiv 21\pmod{101} \implies x^2+x+1\equiv 21^2+ 21+1\equiv 59\pmod{101}$
$x\equiv 22\pmod{101} \implies x^2+x+1\equiv 22^2+ 22+1\equiv  2\pmod{101}$
$x\equiv 23\pmod{101} \implies x^2+x+1\equiv 23^2+ 23+1\equiv 48\pmod{101}$
$x\equiv 24\pmod{101} \implies x^2+x+1\equiv 24^2+ 24+1\equiv 96\pmod{101}$
$x\equiv 25\pmod{101} \implies x^2+x+1\equiv 25^2+ 25+1\equiv 45\pmod{101}$
$x\equiv 26\pmod{101} \implies x^2+x+1\equiv 26^2+ 26+1\equiv 97\pmod{101}$
$x\equiv 27\pmod{101} \implies x^2+x+1\equiv 27^2+ 27+1\equiv 50\pmod{101}$
$x\equiv 28\pmod{101} \implies x^2+x+1\equiv 28^2+ 28+1\equiv  5\pmod{101}$
$x\equiv 29\pmod{101} \implies x^2+x+1\equiv 29^2+ 29+1\equiv 63\pmod{101}$
$x\equiv 30\pmod{101} \implies x^2+x+1\equiv 30^2+ 30+1\equiv 22\pmod{101}$
$x\equiv 31\pmod{101} \implies x^2+x+1\equiv 31^2+ 31+1\equiv 84\pmod{101}$
$x\equiv 32\pmod{101} \implies x^2+x+1\equiv 32^2+ 32+1\equiv 47\pmod{101}$
$x\equiv 33\pmod{101} \implies x^2+x+1\equiv 33^2+ 33+1\equiv 12\pmod{101}$
$x\equiv 34\pmod{101} \implies x^2+x+1\equiv 34^2+ 34+1\equiv 80\pmod{101}$
$x\equiv 35\pmod{101} \implies x^2+x+1\equiv 35^2+ 35+1\equiv 49\pmod{101}$
$x\equiv 36\pmod{101} \implies x^2+x+1\equiv 36^2+ 36+1\equiv 20\pmod{101}$
$x\equiv 37\pmod{101} \implies x^2+x+1\equiv 37^2+ 37+1\equiv 94\pmod{101}$
$x\equiv 38\pmod{101} \implies x^2+x+1\equiv 38^2+ 38+1\equiv 69\pmod{101}$
$x\equiv 39\pmod{101} \implies x^2+x+1\equiv 39^2+ 39+1\equiv 46\pmod{101}$
$x\equiv 40\pmod{101} \implies x^2+x+1\equiv 40^2+ 40+1\equiv 25\pmod{101}$
$x\equiv 41\pmod{101} \implies x^2+x+1\equiv 41^2+ 41+1\equiv  6\pmod{101}$
$x\equiv 42\pmod{101} \implies x^2+x+1\equiv 42^2+ 42+1\equiv 90\pmod{101}$
$x\equiv 43\pmod{101} \implies x^2+x+1\equiv 43^2+ 43+1\equiv 75\pmod{101}$
$x\equiv 44\pmod{101} \implies x^2+x+1\equiv 44^2+ 44+1\equiv 62\pmod{101}$
$x\equiv 45\pmod{101} \implies x^2+x+1\equiv 45^2+ 45+1\equiv 51\pmod{101}$
$x\equiv 46\pmod{101} \implies x^2+x+1\equiv 46^2+ 46+1\equiv 42\pmod{101}$
$x\equiv 47\pmod{101} \implies x^2+x+1\equiv 47^2+ 47+1\equiv 35\pmod{101}$
$x\equiv 48\pmod{101} \implies x^2+x+1\equiv 48^2+ 48+1\equiv 30\pmod{101}$
$x\equiv 49\pmod{101} \implies x^2+x+1\equiv 49^2+ 49+1\equiv 27\pmod{101}$
$x\equiv 50\pmod{101} \implies x^2+x+1\equiv 50^2+ 50+1\equiv 26\pmod{101}$
$x\equiv 51\pmod{101} \implies x^2+x+1\equiv 51^2+ 51+1\equiv 27\pmod{101}$
$x\equiv 52\pmod{101} \implies x^2+x+1\equiv 52^2+ 52+1\equiv 30\pmod{101}$
$x\equiv 53\pmod{101} \implies x^2+x+1\equiv 53^2+ 53+1\equiv 35\pmod{101}$
$x\equiv 54\pmod{101} \implies x^2+x+1\equiv 54^2+ 54+1\equiv 42\pmod{101}$
$x\equiv 55\pmod{101} \implies x^2+x+1\equiv 55^2+ 55+1\equiv 51\pmod{101}$
$x\equiv 56\pmod{101} \implies x^2+x+1\equiv 56^2+ 56+1\equiv 62\pmod{101}$
$x\equiv 57\pmod{101} \implies x^2+x+1\equiv 57^2+ 57+1\equiv 75\pmod{101}$
$x\equiv 58\pmod{101} \implies x^2+x+1\equiv 58^2+ 58+1\equiv 90\pmod{101}$
$x\equiv 59\pmod{101} \implies x^2+x+1\equiv 59^2+ 59+1\equiv  6\pmod{101}$
$x\equiv 60\pmod{101} \implies x^2+x+1\equiv 60^2+ 60+1\equiv 25\pmod{101}$
$x\equiv 61\pmod{101} \implies x^2+x+1\equiv 61^2+ 61+1\equiv 46\pmod{101}$
$x\equiv 62\pmod{101} \implies x^2+x+1\equiv 62^2+ 62+1\equiv 69\pmod{101}$
$x\equiv 63\pmod{101} \implies x^2+x+1\equiv 63^2+ 63+1\equiv 94\pmod{101}$
$x\equiv 64\pmod{101} \implies x^2+x+1\equiv 64^2+ 64+1\equiv 20\pmod{101}$
$x\equiv 65\pmod{101} \implies x^2+x+1\equiv 65^2+ 65+1\equiv 49\pmod{101}$
$x\equiv 66\pmod{101} \implies x^2+x+1\equiv 66^2+ 66+1\equiv 80\pmod{101}$
$x\equiv 67\pmod{101} \implies x^2+x+1\equiv 67^2+ 67+1\equiv 12\pmod{101}$
$x\equiv 68\pmod{101} \implies x^2+x+1\equiv 68^2+ 68+1\equiv 47\pmod{101}$
$x\equiv 69\pmod{101} \implies x^2+x+1\equiv 69^2+ 69+1\equiv 84\pmod{101}$
$x\equiv 70\pmod{101} \implies x^2+x+1\equiv 70^2+ 70+1\equiv 22\pmod{101}$
$x\equiv 71\pmod{101} \implies x^2+x+1\equiv 71^2+ 71+1\equiv 63\pmod{101}$
$x\equiv 72\pmod{101} \implies x^2+x+1\equiv 72^2+ 72+1\equiv  5\pmod{101}$
$x\equiv 73\pmod{101} \implies x^2+x+1\equiv 73^2+ 73+1\equiv 50\pmod{101}$
$x\equiv 74\pmod{101} \implies x^2+x+1\equiv 74^2+ 74+1\equiv 97\pmod{101}$
$x\equiv 75\pmod{101} \implies x^2+x+1\equiv 75^2+ 75+1\equiv 45\pmod{101}$
$x\equiv 76\pmod{101} \implies x^2+x+1\equiv 76^2+ 76+1\equiv 96\pmod{101}$
$x\equiv 77\pmod{101} \implies x^2+x+1\equiv 77^2+ 77+1\equiv 48\pmod{101}$
$x\equiv 78\pmod{101} \implies x^2+x+1\equiv 78^2+ 78+1\equiv  2\pmod{101}$
$x\equiv 79\pmod{101} \implies x^2+x+1\equiv 79^2+ 79+1\equiv 59\pmod{101}$
$x\equiv 80\pmod{101} \implies x^2+x+1\equiv 80^2+ 80+1\equiv 17\pmod{101}$
$x\equiv 81\pmod{101} \implies x^2+x+1\equiv 81^2+ 81+1\equiv 78\pmod{101}$
$x\equiv 82\pmod{101} \implies x^2+x+1\equiv 82^2+ 82+1\equiv 40\pmod{101}$
$x\equiv 83\pmod{101} \implies x^2+x+1\equiv 83^2+ 83+1\equiv  4\pmod{101}$
$x\equiv 84\pmod{101} \implies x^2+x+1\equiv 84^2+ 84+1\equiv 71\pmod{101}$
$x\equiv 85\pmod{101} \implies x^2+x+1\equiv 85^2+ 85+1\equiv 39\pmod{101}$
$x\equiv 86\pmod{101} \implies x^2+x+1\equiv 86^2+ 86+1\equiv  9\pmod{101}$
$x\equiv 87\pmod{101} \implies x^2+x+1\equiv 87^2+ 87+1\equiv 82\pmod{101}$
$x\equiv 88\pmod{101} \implies x^2+x+1\equiv 88^2+ 88+1\equiv 56\pmod{101}$
$x\equiv 89\pmod{101} \implies x^2+x+1\equiv 89^2+ 89+1\equiv 32\pmod{101}$
$x\equiv 90\pmod{101} \implies x^2+x+1\equiv 90^2+ 90+1\equiv 10\pmod{101}$
$x\equiv 91\pmod{101} \implies x^2+x+1\equiv 91^2+ 91+1\equiv 91\pmod{101}$
$x\equiv 92\pmod{101} \implies x^2+x+1\equiv 92^2+ 92+1\equiv 73\pmod{101}$
$x\equiv 93\pmod{101} \implies x^2+x+1\equiv 93^2+ 93+1\equiv 57\pmod{101}$
$x\equiv 94\pmod{101} \implies x^2+x+1\equiv 94^2+ 94+1\equiv 43\pmod{101}$
$x\equiv 95\pmod{101} \implies x^2+x+1\equiv 95^2+ 95+1\equiv 31\pmod{101}$
$x\equiv 96\pmod{101} \implies x^2+x+1\equiv 96^2+ 96+1\equiv 21\pmod{101}$
$x\equiv 97\pmod{101} \implies x^2+x+1\equiv 97^2+ 97+1\equiv 13\pmod{101}$
$x\equiv 98\pmod{101} \implies x^2+x+1\equiv 98^2+ 98+1\equiv  7\pmod{101}$
$x\equiv 99\pmod{101} \implies x^2+x+1\equiv 99^2+ 99+1\equiv  3\pmod{101}$
$x\equiv100\pmod{101} \implies x^2+x+1\equiv100^2+100+1\equiv  1\pmod{101}$