What is the unit digit of the expression K = (377^{123} + 428^{164}) \times 533^{200} - 61… — State PSC Exam

What is the unit digit of the expression $K = (377^{123} + 428^{164}) \times 533^{200} - 619^{86}$ ?

Cevap: 8

Cevap

The unit digit of the expression is 8.

Using cyclicity principles for each power term,

377^{123}

yields unit digit 3 (remainder 3),

428^{164}

yields unit digit 6 (remainder 0

\rightarrow

4th power),

533^{200}

yields unit digit 1 (remainder 0

\rightarrow

4th power), and

619^{86}

yields unit digit 1 (even power of 9). Combining these gives

(3 + 6) \times 1 - 1 = 8

Adım Adım Çözüm

Find the unit digit of

377^{123}

Unit digit is 3

The unit digit of 377 is 7, which follows a cyclicity pattern of 4 (7, 9, 3, 1). Dividing the exponent 123 by 4 leaves a remainder of 3. Thus, the unit digit is given by

7^3 = 343 \rightarrow 3

Find the unit digit of

428^{164}

Unit digit is 6

The unit digit of 428 is 8, which has a cyclicity of 4 (8, 4, 2, 6). The exponent 164 is divisible by 4 (remainder 0), which corresponds to the 4th position in the cyclicity cycle (

8^4 = 4096 \rightarrow 6

Evaluate the sum inside the parentheses

(377^{123} + 428^{164})

Unit digit is 9

Adding the individual unit digits gives

3 + 6 = 9

Find the unit digit of

533^{200}

Unit digit is 1

The unit digit of 533 is 3, which has a cyclicity of 4 (3, 9, 7, 1). Exponent 200 is divisible by 4 (remainder 0), corresponding to the 4th power in the cycle (

3^4 = 81 \rightarrow 1

Multiply the bracketed sum by

533^{200}

Unit digit is 9

Multiplying the unit digits yields

9 \times 1 = 9

Find the unit digit of

619^{86}

Unit digit is 1

The unit digit of 619 is 9, which has a cyclicity of 2 (9 for odd powers, 1 for even powers). Since 86 is even, the unit digit is 1.

Subtract

619^{86}

from the product

Unit digit is 8

Subtracting the unit digits gives

9 - 1 = 8

Anahtar Kavram

Unit Digit and Cyclicity

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