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2014-05-23T10:00:21+05:30

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There is a shortcut method to solve such problems
Suppose x is raised to the power y and we need to find unit digit of  x^{y} .
So, we will first divide y by 4 and find the remainder thus obtained. Since on dividing and integral number by four can yield only four remainders --> 0, 1, 2 and 3.
This method applies for 4 values of x only, because for other values, you can split the values in the 4 values of x:-
x = 2, 3, 7 and 8.
For 2
====
If remainder [y/4] is 1, unit digit of   2^{y} = 2
If remainder is 2, unit digit of  2^{y} = 4
If remainder is 3, unit digit of  2^{y} = 8
If remainder is 0, unit digit of  2^{y} = 6

For 3
====
If remainder is 1, unit digit of  3^{y} = 3
If remainder is 2, unit digit of  3^{y} = 9
If remainder is 3, unit digit of  3^{y} = 7
If remainder is 0, unit digit of  3^{y} = 1

For 7
====
If remainder is 1, unit digit of  7^{y} = 7
If remainder is 2, unit digit of  7^{y} = 9
If remainder is 3, unit digit of  7^{y} = 3
If remainder is 0, unit digit of  7^{y} = 1

For 8
====
If remainder is 1, unit digit of  8^{y} = 8
If remainder is 2, unit digit of  8^{y} = 4
If remainder is 3, unit digit of  8^{y} = 2
If remainder is 4, unit digit of  8^{y} = 6

For memorising that of 3 and 7, you can check the unit digit of when 3 and 7 are raised to the power remainder [power = remainder]. For 2 and 8 this trend doesn't works.
So in your question,
 373^{333}
Dividing 333 by 4, we get remainder 1.
So using table, when remainder is 3, unit digit of 3 [because the unit digit of 373 raised to power something is because of its original unit digit only] will be 3.
Hence unit digit of  373^{333} will be 3.
Hope you understood   :-)
If not please comment.    :-)
2 5 2
Happy to help !
Well you can also memorise as, the unit digit will be power of the remainder, except when the remainder is zero, you should take zero as 4 to calculate the unit digit.