Patterns in Pascal's Triangle mod N
In modular addition, we add two numbers ordinarily and then calculate the remainder of their sum upon division by the given natural number \(N\), called the modulus. For example,
\[1 + 1 = 0 \bmod 2\] \[2 + 3 = 1 \bmod 4\]because 2 leaves a remainder of 0 upon division by 2, and 5 leaves a remainder of 1 upon division by 4. We count the hours on a clock modulo 12; if your watch says 10 o'clock now, then in 7 hours it will be 5 o'clock because
\[10 + 7 = 5 \bmod 12\]With the help of modular addition, we can continue adding rows to Pascal's triangle without generating unreasonably large numbers. In fact, we can calculate the entries in Pascal's triangle modulo any natural number N we wish, and study the resulting patterns.
Let's calculate the entries in the rows 0 through 8 of Pascal's triangle for various moduli. Do you see any indications of patterns in these triangles?
