If the unit digit of $13^n + 7^n - 3^n$, $n \in \mathbb{N}$, is 3 then possible value(s) of $n$ is/are :
Step-by-Step Solution
Key Concept: The unit digit of the expression depends only on $n \bmod 4$, and we need $n \equiv 3 \pmod{4}$ to get unit digit 3.
We need to find the unit digit of $13^n + 7^n - 3^n$ and determine when it equals 3. The unit digits of powers follow cycles: $13^n$ has cycle $(3,9,7,1)$ with period 4, $7^n$ has cycle $(7,9,3,1)$ with period 4, and $3^n$ has cycle $(3,9,7,1)$ with period 4. For each $n \bmod 4$: when $n \equiv 1 \pmod{4}$: unit digit is $3+7-3=7$; when $n \equiv 3 \pmod{4}$: unit digit is $7+3-7=3$. So we need $n \equiv 3 \pmod{4}$. Checking options: $27 \equiv 3 \pmod{4}$ ✓, $103 \equiv 3 \pmod{4}$ ✓, $11 \equiv 3 \pmod{4}$ ✓, $101 \equiv 1 \pmod{4}$ ✗.
Correct Answer: 1,2,3