What is the unit digit of the composite expression E = (648^{116} \times 437^{85}) - 322^{… — State PSC Exam

What is the unit digit of the composite expression $E = (648^{116} \times 437^{85}) - 322^{90}$ ?

The unit digit of the expression is 8.

The term

648^{116}

has a unit digit of 6 (since

116 \pmod 4 = 0

, giving

8^4 \equiv 6 \pmod{10}

), and

437^{85}

has a unit digit of 7 (since

85 \pmod 4 = 1

, giving

7^1 \equiv 7 \pmod{10}

). The product of these two terms yields a unit digit of

(6 \times 7) \pmod{10} = 2

. The subtracted term

322^{90}

has a unit digit of 4 (since

90 \pmod 4 = 2

, giving

2^2 \equiv 4 \pmod{10}

). Subtracting gives

2 - 4 = -2 \equiv 8 \pmod{10}

Find the unit digit of

648^{116}

Unit digit is 6

The base unit digit is 8, which follows a cyclicity of 4 (8, 4, 2, 6). Exponent

116 \div 4 = 29

with remainder 0. A remainder of 0 corresponds to the 4th position in the cyclicity cycle (

8^4 = 4096

), giving a unit digit of 6.

Find the unit digit of

437^{85}

Unit digit is 7

The base unit digit is 7, which follows a cyclicity of 4 (7, 9, 3, 1). Exponent

85 \div 4 = 21

with remainder 1. A remainder of 1 corresponds to the 1st power (

7^1 = 7

), giving a unit digit of 7.

Calculate the unit digit of the product

(648^{116} \times 437^{85})

Unit digit is 2

Multiplying the unit digits from Step 1 and Step 2 gives

6 \times 7 = 42

, which has a unit digit of 2.

Find the unit digit of

322^{90}

Unit digit is 4

The base unit digit is 2, which follows a cyclicity of 4 (2, 4, 8, 6). Exponent

90 \div 4 = 22

with remainder 2. A remainder of 2 corresponds to the 2nd power (

2^2 = 4

), giving a unit digit of 4.

Subtract to find the overall unit digit of

E

Unit digit is 8

Subtracting the unit digits gives

2 - 4 = -2

. Converting to a positive unit digit in base 10 arithmetic gives

-2 + 10 = 8

Unit Digit and Cyclicity Rules for Exponential Expressions

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