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 determine the units digit of a power of 7, we examine the cyclical pattern of the units digits. The units digits of successive powers of 7 are as follows:
\[ 7^1 = 7 \quad (\text{units digit } 7) \\ 7^2 = 49 \quad (\text{units digit } 9) \\ 7^3 = 343 \quad (\text{units digit } 3) \\ 7^4 = 2401 \quad (\text{units digit } 1) \]
The sequence of units digits (7, 9, 3, 1) repeats every 4 powers.
To find the units digit of \( 7^{295} \), we determine the position within this cycle by calculating \( 295 \mod 4 \):
\[ 295 \div 4 = 73 \text{ with a remainder of } 3 \Rightarrow 295 \equiv 3 \pmod{4} \]
Therefore, \( 7^{295} \) has the same units digit as \( 7^3 \), which is \( \boxed{3} \).