We must find a pattern in the powers of 3:
[tex] 3^0 = 1 [/tex]
[tex] 3^1 = 3 [/tex]
[tex] 3^2 = 9 [/tex]
[tex] 3^3 = 27 [/tex]
[tex] 3^4 = 81 [/tex]
So, as you can see, both [tex] 3^0 [/tex] and [tex] 3^4 [/tex] end with 1. So, we can generalize as follows:
[tex] 3^{4k+0} \text{ ends with } 1 [/tex]
[tex] 3^{4k+1} \text{ ends with } 3 [/tex]
[tex] 3^{4k+2} \text{ ends with } 9 [/tex]
[tex] 3^{4k+3} \text{ ends with } 7 [/tex]
Since [tex] 58 = 14\cdot 4 + 2 [/tex], we can deduce that [tex] 3^{58} [/tex] ends with 9.
