What is the rightmost digit preceding the zeros in the value of 30^30?
- (a) 1
- (b) 3
- (c) 7
- (d) 9 ✓ UPSC's answer
Why the answer is (d)
• The number $30^{30}$ can be expressed as $(3 \times 10)^{30} = 3^{30} \times 10^{30}$.
• The term $10^{30}$ contributes exactly 30 trailing zeros to the number.
• The 'rightmost digit preceding the zeros' is therefore the units digit of $3^{30}$.
• The units digits of powers of 3 follow a repeating cycle of 4: 3, 9, 7, 1.
• To find the units digit of $3^{30}$, divide the exponent 30 by the cycle length 4: $30 \div 4 = 7$ with a remainder of 2.
• The remainder 2 corresponds to the second number in the cycle, which is 9.
Why the other options are wrong
- (a) 1
- Option (a) is incorrect because the remainder of 30 divided by 4 is 2, not 4 (or 0), so the units digit is 9, not 1.
- (b) 3
- Option (b) is incorrect because the remainder of 30 divided by 4 is 2, not 1, so the units digit is 9, not 3.
- (c) 7
- Option (c) is incorrect because the remainder of 30 divided by 4 is 2, not 3, so the units digit is 9, not 7.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.