What is the unit digit of the expression E = (842^{324} \times 617^{153}) + 358^{217}? — State PSC Exam

What is the unit digit of the expression $E = (842^{324} \times 617^{153}) + 358^{217}$ ?

Answer: 0

The unit digit of the expression

E = (842^{324} \times 617^{153}) + 358^{217}

is 0.

For

842^{324}

, exponent

324 \pmod 4 = 0

corresponds to the 4th power position in cyclicity (

2^4 = 16 \rightarrow 6

). For

617^{153}

153 \pmod 4 = 1

corresponds to

7^1 = 7

. The product of these terms ends in

6 \times 7 = 42 \rightarrow 2

. For

358^{217}

217 \pmod 4 = 1

corresponds to

8^1 = 8

. Adding the product's unit digit (2) and

358^{217}

's unit digit (8) gives

2 + 8 = 10 \rightarrow 0

Find the unit digit of

842^{324}

Unit digit is 6

The unit digit of base 842 is 2, which repeats unit digits in a cycle of 4:

(2, 4, 8, 6)

. Dividing exponent 324 by 4 yields remainder 0, indicating the 4th position in the cycle,

2^4 = 16 \rightarrow 6

Find the unit digit of

617^{153}

Unit digit is 7

The unit digit of base 617 is 7, which repeats in a cycle of 4:

(7, 9, 3, 1)

. Dividing exponent 153 by 4 yields remainder 1, indicating the 1st position in the cycle,

7^1 = 7

Calculate the unit digit of the product

(842^{324} \times 617^{153})

Unit digit is 2

Multiplying the unit digits of the two terms gives

6 \times 7 = 42

, which has a unit digit of 2.

Find the unit digit of

358^{217}

Unit digit is 8

The unit digit of base 358 is 8, which repeats in a cycle of 4:

(8, 4, 2, 6)

. Dividing exponent 217 by 4 yields remainder 1, indicating the 1st position in the cycle,

8^1 = 8

Add the unit digit of the product to the unit digit of

358^{217}

Final unit digit is 0

Adding the unit digit of the product (2) and the unit digit of the final term (8) gives

2 + 8 = 10

, which ends in 0.

Unit Digit Computation using Exponents and Cyclicity

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