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Prove by mathematical induction that [tex]\:3^{4n}-1[/tex] is divisible by 80.

Verify [tex]S(1)\!:\;3^4-1 \:=\:80\;\text{ . . . True!}[/tex]

Assume [tex]S(k)\!:\;3^{4k}- 1 \;=\;80a\,\text{ for some integer }a.[/tex]

Add [tex]80\!\cdot\!3^{4k}[/tex] to both sides.

$\qquad 3^{4k}-1 + 80\!\cdot\!3^{4k} \;=\;80a + 80\!\cdot\!3^{4k}$

$\qquad 3^{4k} - 1 + (3^4-1)3^{4k} \;=\;80(a + 3^{4k}) $

$\qquad 3^{4k} - 1 + 3^{4k+4} - 3^{4k} \;=\;80(a+3^{4k})$

$\qquad 3^{4(k+1)} - 1 \;=\;80b\;\text{ for some integer }b$

And we have proved $S(k+1).$

The inductive proof is complete.