Divisibility properties of binomial coefficients

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m . For the case when m equals a prime number, p , the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p ) was completely described and the following result (stated by G. Gilbart-Smith) was proved: In the ( n + l)th row of Pascal’s triangle, there are