I've read all the previous posts and it seems one step is missing (certainly too obvious for most of you guys) but for the others, check it out (I've eluded the parenthesis to simplify):

333^222=3^222 x 111^222

111^222 / 7 is found by actually dividing 111 by 7;

=>111=77+41=77+42-1

111 =77+35+6=77+35+7-1

=>with the remainder -1 comes -1^222=1, because 222 is even

3^222 =9^111 gives the following remainders when divided by 7:

9=7-2 => 2^111

8=7+1=> 1^37 (111 is a multiple of 3 as the sum of its digits is a multiple of 3, you don't need to find the quotient 37, you only to have 111=3x, to get 2^3=8)

=> Thus the remainder of this part is 1

Finally, the remainder for the whole number is 1x1 =1

Hope it'll help

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