What is the unit digit of the composite exponential expression E = 433^{45} + 818^{80} - 6… — State PSC Exam

What is the unit digit of the composite exponential expression $E = 433^{45} + 818^{80} - 625^{75}$ ?

Evaluating each term individually:

433^{45}

has unit digit

3

(since

45 \pmod 4 = 1

818^{80}

has unit digit

6

(since

80 \pmod 4 = 0

, giving the 4th power unit digit

8^4 \rightarrow 6

), and

625^{75}

has unit digit

5

. Combining these gives

3 + 6 - 5 = 4

Find the unit digit of

433^{45}

Unit digit is 3

The unit digit of base 433 is 3. The cyclicity of 3 is 4 (

3^1=3, 3^2=9, 3^3=7, 3^4=1

). Dividing the exponent 45 by 4 gives

45 = 4 \times 11 + 1

(remainder 1). Thus, the unit digit is

3^1 = 3

Find the unit digit of

818^{80}

Unit digit is 6

The unit digit of base 818 is 8. The cyclicity of 8 is 4 (

8^1=8, 8^2=4, 8^3=2, 8^4=6

). Dividing exponent 80 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power, giving unit digit

6

(from

8^4 = 4096

Find the unit digit of

625^{75}

Unit digit is 5

The unit digit of base 625 is 5. Any positive integer power of a number ending in 5 always ends in 5 (cyclicity of 1).

Combine the unit digits of all terms

Substitute the individual unit digits into the expression:

3 + 6 - 5 = 4

Unit Digit and Cyclicity Rules for Exponential Expressions

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