What is the unit digit in the expansion of (57242)^(9 × 7 × 5 × 3 × 1)?

Correct Answer: (A) 2

The unit digit of 57242 is 2. The powers of 2 have a cycle of 4 in their units digits: $2^1 \to 2$, $2^2 \to 4$, $2^3 \to 8$, $2^4 \to 6$. The exponent is $E = 9$$\times$ 7 $\times$ 5 $\times$ 3 $\times$ 1. Evaluating $E$ modulo 4: $9$$\equiv$ 1, $7$$\equiv$ 3, $5$$\equiv$ 1, and $3$$\equiv$ 3, so $E$$\equiv$ 1 $\times$ 3 $\times$ 1 $\times$ 3 = 9 $\equiv$ 1 \pmod 4. Since the exponent is congruent to 1 mod 4, the unit digit is $2^1 = 2$.