Let's explore Pascal's Triangle first in terms of mod 2. This means that every entry is divided by 2, and replaced with the value of the remainder. So, the original 10 rows we were looking at before will now consist of either 0's or 1's. We know that every even number is divisible by 2, and will therefore be represented as a 0. Similarly, every odd number is divisible by 2 with a remainder of 1, and that will give us the 1 values. So, when we express Pascal's Triangle in mod 2, we are really looking at the pattern of entries that are odd, and those that are even.
Notice our original ten rows below.
Now, let's associate a color with the corresponding entries, to better identify a pattern. Here, even numbers are left white and odd numbers are shown in red. Fortunately, Microsoft Excel has a clever tool to assign colors to certain cell values, to quickly generate our images.
Now, expand our triangle to the first 63 rows! Watch as our pattern grows. The white space represents all the numbers of Pascal's triangle that are even! Notice how the pattern of the smaller red triangle, made in the corner with the first 15 rows, has just been repeated at a larger scale. Can you imagine what the pattern would look like if we had not stopped after 63 rows?
For comparison, mod 4 was also explored as compared to mod 2. Both modulus values are even, however dividing all of the entries by 4 no longer separates the entries into even and odd numbers. However, the patterns of both are somewhat similar!
Click here to compare mod 2 to mod 4!