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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
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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
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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 $b$)?

see:  https://web2.0calc.com/questions/help-please_33660#r2

 

laugh

 May 28, 2019

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