Proving that a number is composite I have proved that the number $10^{5}2^{17}+1$ is composite by showing that it is divisible by 3 , using remainders. I want an alternative proof.I am looking for a very elementary proof that does not mention remainders. 

$10^5 \pmod 3 = (1)^5 \pmod 3 = 1 \pmod 3$.
$2^{17} \pmod 3 = (-1)^{17} \pmod 3 = -1 \pmod 3$.
Thus, $10^5 \times 2^{17} \pmod 3 = 1 \times (-1) \pmod 3 = -1 \pmod 3$.
Now, $10^5 \times 2^{17} + 1 \pmod 3 = -1 + 1 \pmod 3 = 0 \pmod 3$.
