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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 $b$)?

May 27, 2019
edited by Guest  May 27, 2019

#1
0

1 - 7^2010 mod 100 = 49

May 27, 2019
#2
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thanks but what about the second problem?

May 27, 2019
edited by Guest  May 27, 2019
#3
+24389
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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 $b$)?

May 28, 2019