For which n is the term an integer & Calculate the equivalence

[mathmari]

Hey!

Question 1: We consider $\frac{2n-1}{n+7}$. For which $n$ is this term an integer? I have done the following:

We set $n+7=m \Rightarrow n=m-7$.

Then we get $$\frac{2n-1}{n+7}=\frac{2(m-7)-1}{(m-7)+7}=\frac{2m-15}{m}$$ So $m$ has to be a divisor of $15$, i.e. $m\in \{1,3,5,15\}$, therefore $n\in \{-6, \ -4, \ -2, \ 8\}$.
Question 2: Calculate $12673^{37}\pmod 5$. I have done the following:

From Euler's theorem we have $x^4\equiv 1\pmod 5$.

Then we get \begin{align*}12673^{9\cdot 4+1}\pmod 5&\equiv \left (12673^{4}\right )^9\cdot 12673 \pmod 5\\ & \equiv 1^9\cdot 12673 \pmod 5\\ & \equiv 12673 \pmod 5\\ & \equiv \left (2534\cdot 5+3\right )\pmod 5\\ & \equiv 3\pmod 5\end{align*}
Is everything correct and complete? :unsure:

 

[kaliprasad]

Question 2 is correct.

In question 1 we need to take -1, -3, -5, -15 as additional values of m

 

[mathmari]

Question 2 is correct.

In question 1 we need to take -1, -3, -5, -15 as additional values of m

Ah yes! Except from that everything else is correct and compelete, right?

 

[kaliprasad]

yes

 

[mathmari]

yes

Great! Thank you! ☺