If 3²⁰¹⁹ is divided by 10, then what is the remainder?

To find the remainder when 3^2019 is divided by 10, we need to find the unit digit of 3^2019.

Let's observe the pattern of unit digits of powers of 3: 3^1 = 3 (Unit digit is 3) 3^2 = 9 (Unit digit is 9) 3^3 = 27 (Unit digit is 7) 3^4 = 81 (Unit digit is 1) 3^5 = 243 (Unit digit is 3)

The unit digits repeat in a cycle of 4: (3, 9, 7, 1).

To find the unit digit of 3^2019, we divide the exponent (2019) by the length of the cycle (4) and look at the remainder. 2019 ÷ 4 = 504 with a remainder of 3.

Since the remainder is 3, the unit digit of 3^2019 will be the same as the 3rd unit digit in the cycle, which is 7 (from 3^3).

When any number is divided by 10, the remainder is its unit digit. Therefore, when 3^2019 is divided by 10, the remainder is 7.

Analyzing the options: A) 1: This would be the remainder if the exponent left a remainder of 0 (or 4) when divided by 4. B) 3: This would be the remainder if the exponent left a remainder of 1 when divided by 4. C) 7: This is the correct remainder as the exponent 2019 leaves a remainder of 3 when divided by 4. D) 9: This would be the remainder if the exponent left a remainder of 2 when divided by 4.

The final answer is C.