What is the unit digit of the expression P = (678^{120} + 359^{45}) \times 243^{37} - 714^… — State PSC Exam

What is the unit digit of the expression $P = (678^{120} + 359^{45}) \times 243^{37} - 714^{82}$ ?

Answer: 9

The unit digit of the expression is 9.

Evaluating each term using cyclicity rules gives

678^{120} \rightarrow 6

359^{45} \rightarrow 9

243^{37} \rightarrow 3

, and

714^{82} \rightarrow 6

. Combining these gives

(6 + 9) \times 3 - 6 \equiv 15 \times 3 - 6 \equiv 5 \times 3 - 6 \equiv 15 - 6 \equiv 9 \pmod{10}

Calculate the unit digit of

678^{120}

Unit digit is 6

The cyclicity of 8 is 4 (8, 4, 2, 6). Exponent 120 mod 4 = 0, which corresponds to the 4th term in the cycle.

Calculate the unit digit of

359^{45}

Unit digit is 9

The cyclicity of 9 is 2 (9 for odd powers, 1 for even powers). Exponent 45 is odd.

Find the unit digit of

(678^{120} + 359^{45})

Unit digit is 5

Sum of unit digits is 6 + 9 = 15, which has unit digit 5.

Calculate the unit digit of

243^{37}

Unit digit is 3

The cyclicity of 3 is 4 (3, 9, 7, 1). Exponent 37 mod 4 = 1, so the unit digit is 3^1 = 3.

Multiply the sum by

243^{37}

Unit digit is 5

Product of unit digits is 5 * 3 = 15, which has unit digit 5.

Calculate the unit digit of

714^{82}

Unit digit is 6

The cyclicity of 4 is 2 (4 for odd powers, 6 for even powers). Exponent 82 is even.

Compute the final unit digit of the overall expression

Unit digit is 9

Subtracting unit digits yields 5 - 6. Adding 10 for borrowing gives 15 - 6 = 9.

Unit Digit and Cyclicity Rules for Exponential Expressions

Estimated Time:1m 30s

Question