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.