Question

The digit in units place in (259)^59

Answer 1

first let us find the powers for 259 in order and its unit place.
(259)^1 = 259 = 9 (unit place)
(259)^2 = 67081 = 1
(259)^3 = 17373979 = 9
(259)^4 = 4499860561 = 1
So the unit place comes in the order 1,9,1,9......
So it is in a form of cycle with 2 alternatives.
So when x is a multiple of 2 , (259)^n = 1
Since 59 is not a multiple of 2 (259)^59 the unit place is not equal to 1 and it is equal to 9

Answer 2

You may try this if you're familiar with binomial theorem
$259^{59}=(260-1)^{59}=10M+(-1)^{59}=10M-1=10(M-1)+9$

Therefore the units place in $259^{59}$ is $\boxed{9}$