Find the unit digit of the numerical expression N = (567^{63} \times 234^{45}) - 343^{37}. — State PSC Exam

Find the unit digit of the numerical expression $N = (567^{63} \times 234^{45}) - 343^{37}$ .

Answer: 9

The unit digit of

567^{63}

is 3 and for

234^{45}

it is 4, making their product's unit digit 2. The unit digit of

343^{37}

is 3. Subtracting 3 from 2 with regrouping (

12 - 3

) yields 9.

Determine the unit digit of

567^{63}

Base unit digit is 7 with cyclicity 4. Since

63 \pmod 4 = 3

7^3

gives a unit digit of 3.

Determine the unit digit of

234^{45}

Base unit digit is 4 with cyclicity 2. An odd exponent yields a unit digit of 4.

Multiply the unit digits of the first two terms

The unit digit of the product is

(3 \times 4) \pmod{10} = 2

Determine the unit digit of

343^{37}

Base unit digit is 3 with cyclicity 4. Since

37 \pmod 4 = 1

3^1

gives a unit digit of 3.

Subtract the unit digit of the second part from the first part

Subtracting 3 from 2 requires borrowing 10 (

12 - 3 = 9

) to yield a valid positive unit digit.

Unit Digit and Cyclicity of Powers

Question