Hi Sam,
You have probably seen our response to Landon's question which concerns the last digit of a large power of 7. The technique is very much the same for determining the last two digits of a large power and we will illustrate with 7^{1994}.
The list below contains the last TWO digits for the first 4 powers of 7. This
list repeats as the last two digits through all powers of 7.
07
49
43
01
This repeating list has a period of 4.
1994 = 498 4 + 2
The 2nd term in the list above is 49.
Therefore, the last two digits of 7^{1994} is 49
Now try this technique for 3^{1994}. In this case you will have a much longer list before you find a repeat.
Paul and Penny
