Which one among the following digits can never be at the units place in $(273)^n$, where $n$ is a positive integer?
1
3
5
7
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CISF-AC-EXE – 2024
Let’s examine the pattern of the units digits of powers of 3:
$3^1$: Units digit is 3.
$3^2$: Units digit of $3 \times 3 = 9$.
$3^3$: Units digit of $9 \times 3 = 27$, which is 7.
$3^4$: Units digit of $7 \times 3 = 21$, which is 1.
$3^5$: Units digit of $1 \times 3 = 3$. (The pattern repeats)
$3^6$: Units digit of $3 \times 3 = 9$.
The cycle of the units digits of powers of 3 is 3, 9, 7, 1. This cycle has a length of 4.
For any positive integer n, the units digit of $3^n$ will be one of these four digits: 1, 3, 7, or 9.
Let’s check the given options:
A) 1: Possible (e.g., for $n=4$)
B) 3: Possible (e.g., for $n=1$)
C) 5: Not in the cycle {3, 9, 7, 1}
D) 7: Possible (e.g., for $n=3$)
Therefore, the digit 5 can never be at the units place in $(273)^n$ for any positive integer n.