Omega Numbers

     A popular problem in mathematics classes about problem solving concerns finding the unit's digit of a large power of a number. An example of this might be:

Find the unit's digit of 2⁴⁰⁰⁰.

     Of course, the student solving this is not expected to compute 2 used as a factor 4000 times. The reasons should be obvious. Rather the solver begins by looking for patterns, and armed with that knowledge, deduce the answer in a simple, straight-forward manner.

     We propose now the following variation on this theme:

State the 2-digit number formed
by the final pair of digits of 2²⁰⁰⁴.

     Explain your process clearly, with enough data to establish your claim.

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     Please note: use of a simple calculator (with 8- or 10-digit displays) is permitted, however, such computing aid is not really even necessary. What is not permitted is the use of high-powered computing software, such as Mathematica.