Question 262985: find the unit digit of 3^2002 - 2^2002
a 1 b 3 c 5 d 7
Found 2 solutions by ajmisra, Alan3354:
Answer by ajmisra(4)   (Show Source):
You can put this solution on YOUR website! It pretty easy mate!
3^2000 - 2^2000 can be written as
3^2000 - (3-1)^2000
now open the bracket by distributive law of indices i.e.
(a-b)^x = a^x - b^x
thus, 3^2000 - {3^2000 - 1^2000}
i.e. opening the braces:
3^2000 - 3^2000 + 1^2000 (check that opening the braces causes - and - to become +)
i.e. 1^2000
Now 1^anything is always 1
hence your answer is 1 (i.e. option "a")
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! 3^2002 --> 9 as the 1's digit
Powers of 3 go:
3
9
27
81
243
729
2187
6561
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The units go 1,3,9,7,1,3,9,7...
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In a similar manner, powers of 2 give:
2,4,8,6,2,4,8,6... so the units digit is 4
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9 -4 = 5
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The other tutor's approach would work, I think, but he used ^2000 instead of ^2002.
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Actually, the other method does not work. (a-b)^x is not a^x - b^x
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