Question

the last digit of the number (373)³³³ will be?
[373 to the power 333]

Answer 1

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.    :-)