What is the unit digit of the composite exponential expression E = 332^{520} \times 553^{2… — State PSC Exam

What is the unit digit of the composite exponential expression $E = 332^{520} \times 553^{213} - 448^{631}$ ?

A
1
B
0
6Cevap
D
4

Cevap

The unit digit of the given expression is 6.

Evaluating each component using cyclicity of unit digits:

332^{520}

has base unit digit 2 and exponent divisible by 4 (

520 \bmod 4 = 0

), so its unit digit is

2^4 \implies 6

553^{213}

has base unit digit 3 with

213 \bmod 4 = 1

, giving unit digit

3^1 = 3

448^{631}

has base unit digit 8 with

631 \bmod 4 = 3

, giving unit digit

8^3 \implies 2

. Combining these gives

(6 \times 3) - 2 = 18 - 2 = 16

, which yields a final unit digit of 6.

Adım Adım Çözüm

Determine the unit digit of

332^{520}

using cyclicity of 2

Unit digit is 6

The unit digit of base 332 is 2. The cyclicity of 2 is 4 (2, 4, 8, 6). Dividing the exponent 520 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power:

2^4 = 16

, so the unit digit is 6.

Determine the unit digit of

553^{213}

using cyclicity of 3

Unit digit is 3

The unit digit of base 553 is 3. The cyclicity of 3 is 4 (3, 9, 7, 1). Dividing the exponent 213 by 4 gives a remainder of 1 (

213 = 4 \times 53 + 1

). Thus, the unit digit is

3^1 = 3

Determine the unit digit of

448^{631}

using cyclicity of 8

Unit digit is 2

The unit digit of base 448 is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 631 by 4 gives a remainder of 3 (

631 = 4 \times 157 + 3

). Thus, the unit digit is

8^3 = 512

, which ends in 2.

Combine the unit digits according to the expression

E = (332^{520} \times 553^{213}) - 448^{631}

Unit digit is 6

First multiply the unit digits of the first two terms:

6 \times 3 = 18 \implies 8

. Then subtract the unit digit of the third term:

8 - 2 = 6

Anahtar Kavram

Unit Digit Cyclicity Rules

Tahmini Süre:1m 30s

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