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

1. A. 2
2. B. 4
3. C. 6
4. D. 8

Answer: A

Explanation

To find the unit digit of a number raised to a power, we only need to consider the unit digit of the base number and the cyclicity of its powers. The unit digit of the base number (57242) is 2. The exponent is 9 × 7 × 5 × 3 × 1 = 945. The cyclicity of the unit digit of powers of 2 is: 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16 (unit digit 6), 2^5 = 32 (unit digit 2). The cycle length is 4 (2, 4, 8, 6). To find the unit digit, we divide the exponent (945) by the cycle length (4) and use the remainder. 945 ÷ 4 gives a remainder of 1 (since 945 = 4 × 236 + 1). Therefore, the unit digit of (57242)^945 will be the same as the unit digit of 2^1, which is 2. This question assesses knowledge of number properties, specifically unit digits and cyclicity, a common topic in CSAT.