The last digit (unit digit) of 12^25 is

The correct answer is Option (2) → 2

The unit digit of \( 12^{25} \) is the unit digit of \( 2^{25} \) (working modulo 10). The unit digits of the powers of 2 repeat in a cycle of 4:

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

    \( \displaystyle 25=4\times 6+1 \)

So \( 2^{25} \) ends in the same digit as \( 2^{1} \), and \( 12^{25}\equiv 2\pmod{10} \): the unit digit is 2.