Pascals Triangle

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In 1653, a french mathematician named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or equivalently the number of ways to choose n objects from a group of m indistinguishable objects. The numbers in each row also correspond to the coefficients of (1+x)n

The first few rows of Pascals triangle look like this:

1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

In each row, the two outermost numbers are 1. All others are the sum of the two numbers above (e.g. in row 5, the 6 comes from 3+3).

It turns out that Pascal's triangle holds many interesting numeric patterns. One way of seeing some of these patterns is to pick a number x and color all numbers in the triangle that are evenly divisible by x with one color, and all the other numbers in the triangle with a second color. To see as much of the pattern as possible, you need to be able to see as many rows of the triangle as possible, but coloring a large number of rows like this by hand is very boring and time consuming. A computer can color many rows of the triangle in only a few seconds, so we can use it to look at the results using many different divisors.

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