I realize that your question is about the $100 \times 100$ table: But since you ask about approaching the multiplication table in ways other than by rote learning alone, I thought I would leave you with a list of problems I generated based on the $10 \times 10$ table.
[Edit 5/9/14: You can find some of the problems below in an informal paper of mine; the citation is:
Dickman, B. (2014). Problem Posing with the Multiplication Table. Journal of Mathematics Education at Teachers College, 5(1).
You can also access a free copy online here.]
The inspiration for these problems can be found in a paper by Trivett (1980) in which he writes:
The recommendation here is that the multiplication table should be viewed, apparently for the first time by most people, as a dynamic synergetic combination of patterns, a veritable repository of mathematical relationships waiting as it were to gush forth from kindergarten through the secondary grades (p. 21).
Citation Trivett, J. (1980). The Multiplication Table: To Be Memorized or Mastered? For the Learning of Mathematics, 21-25. Link.
(For A, B, and G, see the concrete example section in my earlier post here.)
A. Estimate how many distinct numbers are in the 100 boxes of a 10x10 multiplication table.
B. How many of the entries in the 10x10 multiplication table are odd?
C. Starting at the 1 in a multiplication table, you can take steps to adjacent boxes: right or left, up or down, but not diagonally. What is the biggest number you can arrive at in 10 steps?
D. Consider the 2x2 sub-grid cutouts from a multiplication table. For which x can you fill in the remaining three numbers with complete surety?
E. Find the only 49 in the multiplication table. Add to it the six numbers above in its column, and the six numbers to the left in its row. What is the total?
F. Add up the numbers from all 100 boxes in a 10x10 multiplication table. What is the total?
G. Using a multiplication table, explain what it means for a number to be “prime.”
H. Look at the numbers on the south-east diagonal starting from either 2 entry in the 10x10 times table: 2, 6, 12, 20, 30, 42, 56, 72, 90. What pattern or patterns do you see?
I. Starting at the 3 in row one of the multiplication table, jump like a knight (from Chess) to move one square down and two to the right. Doing this repeatedly, you get the following numbers: 3, 10, 21, 36. What pattern or patterns do you see?
J. What kinds of symmetry can you find in the 10x10 multiplication table?
K. How would you extend the 10x10 multiplication table to cover the negative numbers?
L. Which numbers are missing from the 10x10 multiplication table and why?
M. How many one syllable number names are found in a 10x10 multiplication table?