1) What is the remainder when $7^{2010}$ is divided by $100$?
2) When the base-$b$ number $11011_b$ is multiplied by $b-1$, then $1001_b$ is added, what is the result (written in base $b$)?
+26400 1)
What is the remainder when \(7^{2010}\) is divided by \(100\) ?
\(\begin{array}{|rcll|} \hline && \mathbf{7^{2010} \pmod{ 100}} \quad & | \quad \mathbf{7^4} \equiv {\color{red}1} \pmod{100} \\ &\equiv & 7^{4\cdot 502+2}\pmod{ 100} \\ &\equiv & \left(\mathbf{7^4}\right)^{502}7^2 \pmod{ 100} \\ &\equiv & \left({\color{red}1}\right)^{502}7^2 \pmod{ 100} \\ &\equiv & 7^2 \pmod{ 100} \\ &\mathbf{\equiv} & \mathbf{49 \pmod{ 100}} \\ \hline \end{array}\)
2)
When the base-$b$ number $11011_b$ is multiplied by $b-1$, then $1001_b$ is added, what is the result (written in base $b$)?
see: https://web2.0calc.com/questions/help-please_33660#r2
