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2015-04-03T22:36:51+05:30

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 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


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2015-04-04T11:18:51+05:30

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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.
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