What is the rightmost digit preceding the zeros in the value of 3030?

Correct Option (D)

To determine the rightmost digit preceding the zeros in $30^{30}$3030, the expression can be factorized as follows:

$30^{30} = (3 \times 10)^{30} = 3^{30} \times 10^{30}$3030=(3×10)30=330×1030

The term $10^{30}$1030 indicates that the number will terminate with 30 zeros. Consequently, the digit immediately preceding these zeros is the unit digit of $3^{30}$330.

The unit digits of powers of 3 follow a repeating cycle:

• Unit digit of $3^{1}$31 is 3
• Unit digit of $3^{2}$32 is 9
• Unit digit of $3^{3}$33 is 7
• Unit digit of $3^{4}$34 is 1

This cycle has a length of 4 (3, 9, 7, 1). To ascertain the unit digit of $3^{30}$330, the exponent 30 is divided by the cycle length 4:

$\frac{30}{4} = 7$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 $3^{30}$330 is 9.

Thus, the rightmost digit preceding the zeros in $30^{30}$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 $3^{30}$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.