Now let's continue our investigation by looking at Pascal's triangle in mod 5. Here, every entry is divided by 5 to produce remainders of 0, 1, 2, 3, or 4. Below is again the first ten rows of Pascal's triangle in mod 5.
Again, the pattern is much easier to see once we add different colors for our remainder values. The 0 values have been left white, and red remains the color of all the 1 entries, and blue the color for 2. Here, we have added yellow to represent the remainder 3, and green for 4.
Now, let's expand our pattern to the 124th row! Here we have to increase the size of our triangle just to see a complete pattern. We can see that now the white space represents values that are multiples of 5. Notice again how the smaller triangle in the upper left corner is repeated at a larger and larger scale throughout the entire triangle. Notice how this pattern occurs regardless of the size of the triangles. What do you think the pattern would look like if we were to expand our entries even further? Other than the triangle color pattern, what other patterns do you notice within each smaller triangle?
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