(7¹⁹ + 2) is divided by 6. The remainder is :

$$\eqalign{ & \left( {{7^{19}} + 2} \right) \div 6 \cr & = \frac{{{{\left( {6 + 1} \right)}^{19}} + 2}}{6} \cr & = \frac{{{1^{19}} + 2}}{6} \cr & = 3\,{\text{remainder}} \cr} $$

Alternate
(xⁿ - aⁿ) is divisible by (x - a) for all values of n.
∴ (7¹⁹ - 1¹⁹) is divisible by (7 - 1)
⇒ (7¹⁹ - 1) is divisible by 6
⇒ On dividing (7¹⁹ + 2) by 6, remainder obtained = 3