Correct Option (D)
To determine the rightmost digit preceding the zeros in 3030, the expression can be factorized as follows:
3030=(3×10)30=330×1030
The term 1030 indicates that the number will terminate with 30 zeros. Consequently, the digit immediately preceding these zeros is the unit digit of 330.
The unit digits of powers of 3 follow a repeating cycle:
- Unit digit of 31 is 3
- Unit digit of 32 is 9
- Unit digit of 33 is 7
- Unit digit of 34 is 1
This cycle has a length of 4 (3, 9, 7, 1). To ascertain the unit digit of 330, the exponent 30 is divided by the cycle length 4:
430=7 with a remainder of 2.
The remainder of 2 signifies that the unit digit will be the second digit in the cycle. Therefore, the unit digit of 330 is 9.
Thus, the rightmost digit preceding the zeros in 3030 is 9.
Incorrect Options:
Options A (1), B (3), and C (7) are incorrect as they do not correspond to the unit digit of 330.
- A unit digit of 1 occurs when the exponent, when divided by 4, yields a remainder of 0 (or is a multiple of 4).
- A unit digit of 3 occurs when the exponent, when divided by 4, yields a remainder of 1.
- A unit digit of 7 occurs when the exponent, when divided by 4, yields a remainder of 3.
Since the exponent 30 leaves a remainder of 2 when divided by 4, the correct unit digit is 9, which is the second element in the cycle of unit digits for powers of 3.