Determine the unit digit of the numerical expression N = (333^{444} \times 444^{333}) + (7… — State PSC Exam

Determine the unit digit of the numerical expression $N = (333^{444} \times 444^{333}) + (777^{888} \times 888^{777}) - 222^{555}$ .

Cevap: 4

The unit digit of the given expression is 4.

By analyzing cyclicity of each base (

3, 4, 7, 8, 2

), we find that

333^{444} \rightarrow 1

444^{333} \rightarrow 4

777^{888} \rightarrow 1

888^{777} \rightarrow 8

, and

222^{555} \rightarrow 8

. Thus, the overall unit digit is

(1 \times 4) + (1 \times 8) - 8 = 4 + 8 - 8 = 4

Calculate unit digit of

333^{444} \times 444^{333}

Unit digit is 4

Base 3 has cyclicity 4;

444 \pmod 4 = 0 \implies 3^4 \rightarrow 1

. Base 4 has cyclicity 2; odd exponent

333 \implies 4^1 \rightarrow 4

. Total unit digit for term 1 =

1 \times 4 = 4

Calculate unit digit of

777^{888} \times 888^{777}

Unit digit is 8

Base 7 has cyclicity 4;

888 \pmod 4 = 0 \implies 7^4 \rightarrow 1

. Base 8 has cyclicity 4;

777 \pmod 4 = 1 \implies 8^1 \rightarrow 8

. Total unit digit for term 2 =

1 \times 8 = 8

Calculate unit digit of

222^{555}

Unit digit is 8

Base 2 has cyclicity 4;

555 \pmod 4 = 3 \implies 2^3 \rightarrow 8

Evaluate combined expression unit digit

Combine term unit digits:

(4 + 8 - 8) = 4

Unit Digit and Cyclicity of Numbers

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