Reasoning vs Computation Math Example 2

Solution

1. 1
   Compute units digits of powers of 7: $7^1=7$, $7^2=49$ (units 9), $7^3=343$ (units 3), $7^4=2401$ (units 1), $7^5$ ends in 7.
2. 2
   The units digits cycle with period 4: $7, 9, 3, 1, 7, 9, 3, 1, \ldots$
3. 3
   Reasoning: $100 = 4 \times 25$, so $7^{100}$ is at position $100 \pmod{4} = 0$, which corresponds to the 4th position in the cycle.
4. 4
   The 4th position gives units digit $1$.