# Q. What is the unit digit in the expansion of

(57242)^9×7×5×3×1 ?

(a) 2

(b) 4

(c) 6

(d) 8

Correct Answer: (a) 2

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Unit Digit of a Power **57242**^{945}

^{945}

To find the unit digit of **57242 ^{945}**, 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:

**2**(unit digit is 2)^{1}= 2**2**(unit digit is 4)^{2}= 4**2**(unit digit is 8)^{3}= 8**2**(unit digit is 6)^{4}= 16**2**(unit digit is 2)^{5}= 32**2**(unit digit is 4)^{6}= 64**2**(unit digit is 8)^{7}= 128**2**(unit digit is 6)^{8}= 256**2**(unit digit is 2)^{9}= 512

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 **2 ^{945}**, calculate the remainder of 945 divided by 4:

**945 ÷ 4 = 236 remainder 1**

This remainder tells us that **2 ^{945}** corresponds to the first number in the cycle of unit digits of 2, which is 2.

Thus, the unit digit of **57242 ^{945}** is 2.

Therefore, the correct answer is 2.