What is the unit digit of the expression S = 2^{84} + 4^{97} \times 3^{54} - 9^{45}? — State PSC Exam

What is the unit digit of the expression $S = 2^{84} + 4^{97} \times 3^{54} - 9^{45}$ ?

3Answer
B
8
C
2
D
1

Answer

The unit digit of the expression is 3.

The unit digit of

2^{84}

is 6 (since 84 is a multiple of 4, corresponding to the 4th position in cyclicity). The product

4^{97} \times 3^{54}

has a unit digit of

(4 \times 9) \pmod{10} = 6

. The unit digit of

9^{45}

is 9. Substituting these into the original expression yields

6 + 6 - 9 = 3

Step-by-Step Solution

Find the unit digit of

2^{84}

Unit digit is 6

The cyclicity of

2

4

(

2, 4, 8, 6

). Since

84 \div 4 = 21

with remainder

0

, we take the 4th term in the cyclicity pattern, which is

6

Find the unit digit of

4^{97}

and

3^{54}

Unit digit of

4^{97}

is 4; Unit digit of

3^{54}

is 9

The cyclicity of

4

2

(

4

for odd powers,

6

for even powers). Since

97

is odd,

4^{97}

ends in

4

. The cyclicity of

3

4

(

3, 9, 7, 1

). Since

54 = 4 \times 13 + 2

, remainder is

2

, giving

3^2 = 9

Find the unit digit of the product

4^{97} \times 3^{54}

Unit digit of product is 6

Multiplying the unit digits:

4 \times 9 = 36

, so the unit digit is

6

Find the unit digit of

9^{45}

Unit digit is 9

The cyclicity of

9

2

(

9

for odd powers,

1

for even powers). Since

45

is odd,

9^{45}

ends in

9

Combine the unit digits following the algebraic operations

Unit digit is 3

Evaluating

6 + 6 - 9 = 12 - 9 = 3

Key Concept

Unit Digit and Cyclicity of Exponential Numbers

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