How do I find the units digit of 2012^103

Question

asnaseer · Answer

HINT: $2012=2^2\times503$. Look for patterns in the last digit of $2^n$ and $503^n$ as $n=1,2,...$ then use this information to deduce the result.

anonymous · Answer

still confused can you do a step by step explaination by any chance?

tkhunny · Answer

2 (2)
2^2 = 4 (4)
2^3 = 8 (8)
2^4 = 16 (6)
2^5 = 32 (2)
2^6 = 64 (4)

Do you believe that this pattern might continue?

2, 4, 8, 6, 2, 4, 8, 6, etc.

anonymous · Answer

Yes sorry i was away from my computer

anonymous · Answer

@tkhunny

anonymous · Answer

Can someone please help!

tkhunny · Answer

Did help.  So did asnaseer.  What are your thoughts concerning these hints?

anonymous · Answer

Yes a little.  So I notice the pattern is {2,4,8,16}  So I take the exponent and divide by 4.  I get 25.75.  Then I multiply it by the divisor(4) and I get 100.  Then I subtract 100 from 103 to get 3.  3 is the amount of moves I make around the mod circle which lands on 6.  So the units digit is 6?

anonymous · Answer

|dw:1379310765415:dw|  IS this correct?