What is the unit digit of the expression 7^{43} + 9^{27}? — State PSC Exam

What is the unit digit of the expression $7^{43} + 9^{27}$ ?

Answer: 2

The unit digit of the given expression

7^{43} + 9^{27}

is 2.

The unit digit of

7^{43}

is determined by dividing 43 by 4, giving a remainder of 3, so

7^3

ends in 3. The unit digit of

9^{27}

is determined by taking 9 to an odd power, which ends in 9. Adding

3 + 9 = 12

gives a final unit digit of 2.

Determine the unit digit of

7^{43}

using cyclicity rules.

The unit digit of

7^{43}

is 3.

The base 7 follows a 4-step cyclicity pattern (7, 9, 3, 1). Dividing exponent 43 by 4 leaves remainder 3, giving

7^3 \pmod{10} = 3

Determine the unit digit of

9^{27}

using cyclicity rules.

The unit digit of

9^{27}

is 9.

The base 9 follows a 2-step cyclicity pattern (9 for odd powers, 1 for even powers). Since 27 is odd, the unit digit is 9.

Combine the resulting unit digits.

The unit digit of the sum is 2.

Summing the individual unit digits gives

3 + 9 = 12

, whose unit digit is 2.

Unit Digit and Cyclicity

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