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# help

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Find the largest integer that divides all integers of the ofrm 7^{2n + 1} + 1, for integers n >= 0.

May 18, 2020

#1
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The largest integer appears to be 8. It divides all integers of the form:7^(2n + 1) + 1 from n=1 to infinity.

May 18, 2020
#2
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Find the largest integer that divides all integers of the form  $$7^{2n + 1} + 1$$, for integers $$n \ge 0$$.

My attempt:

$$\text{if }~n=0:~ 7^{2n + 1} + 1 = 7^{2*0 + 1} + 1 =7 + 1 = 8$$

The largest integer that divides $$7^{2n + 1} + 1$$, if $$n=0$$ is $$8$$.

The question is: $$7^{2n + 1} + 1 \equiv 0 \pmod{8} \ ?$$

$$\begin{array}{|rcll|} \hline 7^{2n + 1} + 1 &\equiv& 0 \pmod{8} \ ? \\ \hline && 7^{2n + 1} + 1 \pmod{8} \\ &\equiv& 7^{2n}*7 + 1 \pmod{8} \\ &\equiv& \left(7^2 \right)^n*7 + 1 \pmod{8} \\ &\equiv& {49}^n*7 + 1 \pmod{8} \quad |\quad 49 &\equiv& 1 \pmod{8} \\ &\equiv& {1}^n*7 + 1 \pmod{8} \\ &\equiv& 7 + 1 \pmod{8} \\ &\equiv& 8 \pmod{8} \\ &\equiv& 8-8 \pmod{8} \\ &\equiv& 0 \pmod{8} \\ \hline \end{array}$$

Find the largest integer that divides all integers of the form $$7^{2n + 1} + 1$$ is 8

May 19, 2020