Question

Find the units digit in $7^{295}$.

Hint:

To find the units digit of a large power, identify the cycle of units digits and reduce the exponent modulo the cycle length.

Answer 1

To find the units digit of a power, we observe the repeating pattern in the units digit of powers of 7.
\[ 7^1 = 7 \quad (\text{units digit } 7) \\ 7^2 = 49 \quad (\text{units digit } 9) \\ 7^3 = 343 \quad (\text{units digit } 3) \\ 7^4 = 2401 \quad (\text{units digit } 1) \]
The cycle of units digits is: 7, 9, 3, 1 — and it repeats every 4 powers.
Now, find \( 295 \mod 4 \) to locate the position in the cycle:
\[ 295 \div 4 = 73 \text{ remainder } 3 \Rightarrow 295 \equiv 3 \pmod{4} \]
So, \( 7^{295} \) has the same units digit as \( 7^3 \), which is \( \boxed{3} \).