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

Explanation : 

Unit Digit of a Power 57242⁹⁴⁵

To find the unit digit of 57242⁹⁴⁵, we can focus solely on the unit digit of the base number, which is 2, since the unit digit of a power depends only on the unit digit of the base.

Calculating Power

First, calculate the product 9 x 7 x 5 x 3 x 1 = 945.

Pattern in Powers of 2

Next, observe the pattern in the powers of 2:

• 2¹ = 2 (unit digit is 2)
• 2² = 4 (unit digit is 4)
• 2³ = 8 (unit digit is 8)
• 2⁴ = 16 (unit digit is 6)
• 2⁵ = 32 (unit digit is 2)
• 2⁶ = 64 (unit digit is 4)
• 2⁷ = 128 (unit digit is 8)
• 2⁸ = 256 (unit digit is 6)
• 2⁹ = 512 (unit digit is 2)

The unit digits repeat every four powers (2, 4, 8, 6). This is known as the cyclicity of 2, which is 4.

Finding Relevant Power

To find the relevant power in the cycle for 2⁹⁴⁵, calculate the remainder of 945 divided by 4:

945 ÷ 4 = 236 remainder 1

This remainder tells us that 2⁹⁴⁵ corresponds to the first number in the cycle of unit digits of 2, which is 2.

Thus, the unit digit of 57242⁹⁴⁵ is 2.

Therefore, the correct answer is 2.