The units digit in 12^108 is:

The correct answer is Option (3) → 6

The units digit of \( 12^{n} \) is the units digit of \( 2^{n} \). Powers of 2 have units digits that repeat in a cycle of 4:

    \( \displaystyle 2^{1}\to 2,\ 2^{2}\to 4,\ 2^{3}\to 8,\ 2^{4}\to 6,\ 2^{5}\to 2,\ \ldots \)

108 is divisible by 4 (108 = 4 × 27), so \( 2^{108} \) is in the same position of the cycle as \( 2^{4} \):

    \( \displaystyle 2^{108}=\left(2^{4}\right)^{27}=16^{27} \)

Every power of a number ending in 6 ends in 6. So the units digit of \( 12^{108} \) is 6.

Tip: Divide the exponent by the cycle length (4 for 2, 3, 7, 8). A remainder of 0 means the last digit of the cycle, not the first.