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 ${30}^{30}$, the expression can be factorized as follows:

${30}^{30}=(3\times 10{)}^{30}=3^{30}\times {10}^{30}$

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

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

• Unit digit of $3^1$ is 3
• Unit digit of $3^2$ is 9
• Unit digit of $3^3$ is 7
• Unit digit of $3^4$ is 1

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

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

Thus, the rightmost digit preceding the zeros in ${30}^{30}$ is 9.

Incorrect Options:

Options A (1), B (3), and C (7) are incorrect as they do not correspond to the unit digit of $3^{30}$.

• 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.