What is the unit digit of the expression 432^{216} - 517^{142} - 734^{121} \times 283^{105… — State PSC Exam

What is the unit digit of the expression $432^{216} - 517^{142} - 734^{121} \times 283^{105}$ ?

5Answer
B
0
C
1
D
4

Answer

The unit digit of the given expression is 5.

Evaluating each component:

432^{216}

ends in 6 (since 216 is divisible by 4, giving

2^4 = 16

517^{142}

ends in 9 (

142 \pmod 4 = 2

, giving

7^2 = 49

), and

734^{121} \times 283^{105}

ends in

4 \times 3 = 12 \rightarrow 2

. Combining these gives

(6 - 9 - 2) = -5 \equiv 5 \pmod{10}

. Thus, the option with value 5 is correct.

Step-by-Step Solution

Determine the unit digit of the first term

432^{216}

Unit digit is 6

The base unit digit is 2, which has a cyclicity of 4 (pattern: 2, 4, 8, 6). Dividing exponent 216 by 4 gives a remainder of 0. When the remainder is 0, we use the 4th power:

2^4 = 16

, so the unit digit is 6.

Determine the unit digit of the second term

517^{142}

Unit digit is 9

The base unit digit is 7, which has a cyclicity of 4 (pattern: 7, 9, 3, 1). Dividing exponent 142 by 4 gives a remainder of 2. Taking

7^2 = 49

, the unit digit is 9.

Determine the unit digit of the product term

734^{121} \times 283^{105}

Unit digit is 2

For

734^{121}

, base 4 with an odd exponent gives unit digit 4. For

283^{105}

, base 3 with exponent

105 \pmod 4 = 1

gives unit digit 3. Multiplying their unit digits:

4 \times 3 = 12

, so the unit digit is 2.

Combine the unit digits following order of operations (BODMAS)

Unit digit is 5

We compute

(6 - 9 - 2) \pmod{10}

. Evaluating left-to-right:

6 - 9 = -3 \equiv 7 \pmod{10}

(by adding 10 to handle borrowing). Then

7 - 2 = 5

Key Concept

Unit digit calculation using cyclicity of numbers and modular arithmetic rules under BODMAS

Estimated Time:2m 0s

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