The number 6⁶³⁸, when divided by 13, gives remainder:

10

The correct answer is Option (3) → 10

$6^2 = 36 \equiv 10 \ (\text{mod } 13)$

$6^3 \equiv 10 \cdot 6 = 60 \equiv 8 \ (\text{mod } 13)$

$6^6 \equiv 8^2 = 64 \equiv 12 \equiv -1 \ (\text{mod } 13)$

$6^{12} \equiv 1 \ (\text{mod } 13)$

$638 \equiv 2 \ (\text{mod } 12)$

$6^{638} \equiv 6^2 = 36 \equiv 10 \ (\text{mod } 13)$

$\text{Remainder} = 10$