Question

Use inductive reasoning to determine the units digit of the number 3 to the 58 power

Answer 1

We must find a pattern in the powers of 3:

$3^0 = 1$

$3^1 = 3$

$3^2 = 9$

$3^3 = 27$

$3^4 = 81$

So, as you can see, both $3^0$ and $3^4$ end with 1. So, we can generalize as follows:

$3^{4k+0} \text{ ends with } 1$

$3^{4k+1} \text{ ends with } 3$

$3^{4k+2} \text{ ends with } 9$

$3^{4k+3} \text{ ends with } 7$

Since $58 = 14\cdot 4 + 2$, we can deduce that $3^{58}$ ends with 9.