Question

what is the unit digit of (12) power 53

Answer 1

$12^{53} = (6 * 2)^{53} = 6^{53} 2^{53}$
$6^{53} 2^{52+1} = 6^{53} 2^{52}*2 = 6^{53}2^{4^{13}} 2$
As we know that the unit digit of any power of 6 is six and unit digit of 2 to the power 4 is 6. so we can rewrite above expression as:-
$6^{53}2^{4^{13}} 2 = 6* 6^{13}*2$
Here also unit digit of any power of 6 is 6. So, we come to end by the following expression:-
$6* 6^{13} = 6 * 6 * 2$
Here unit digit of 6*6*2 is 2.
SO THE UNIT DIGIT OF $12^{53}$ is 2

Answer 2

To find the unit digit, find the pattern by checking some smaller powers
12¹ = 12
12² = 144
12³ = 1728
12⁴ = 20736
12⁵ = 248832

12¹ and 12⁵ have same unit digit.
The unit digit started repeating after 4th (=5-1) power.
We need to find the unit digit of 12⁵³.
Remainder of (53 ÷ 4) is 1. So 12⁵³ will have same unit digit as 12¹. So unit digit of 12⁵³ is 2.