Correct Option (C)

To determine the remainder when 3²⁰¹⁹ is divided by 10, one must identify the unit digit of 3²⁰¹⁹.

The pattern of unit digits for successive powers of 3 is as follows:

Unit digit of 3¹ is 3.
Unit digit of 3² is 9.
Unit digit of 3³ is 7 (from 27).
Unit digit of 3⁴ is 1 (from 81).
Unit digit of 3⁵ is 3 (from 243), which indicates the cycle repeats.

The cyclicity of the unit digit of powers of 3 is 4.

To find the unit digit of 3²⁰¹⁹, the exponent 2019 is divided by the cyclicity 4:

2019 ÷ 4 = 504 with a remainder of 3.

Therefore, the unit digit of 3²⁰¹⁹ is equivalent to the unit digit of 3³.

Since 3³ = 27, its unit digit is 7.

When any integer ending in 7 is divided by 10, the remainder is 7.

Incorrect Options:

The remainder when 3ⁿ is divided by 10 is determined by the unit digit of 3ⁿ. The unit digits of powers of 3 follow a repeating cycle of (3, 9, 7, 1), which is dependent on the remainder obtained when the exponent n is divided by 4.

Option (A) 1: This remainder would correspond to a unit digit of 1. This occurs if the exponent 2019 had a remainder of 0 when divided by 4 (i.e., 2019 was a multiple of 4), similar to 3⁴ = 81.
Option (B) 3: This remainder would correspond to a unit digit of 3. This occurs if the exponent 2019 had a remainder of 1 when divided by 4, similar to 3¹ = 3.
Option (D) 9: This remainder would correspond to a unit digit of 9. This occurs if the exponent 2019 had a remainder of 2 when divided by 4, similar to 3² = 9.