MathLabs

Problem 3

Let k,m,nk,m,n be natural numbers such that m+k+1m+k+1 is a prime greater than n+1n+1. Put cs=s(s+1)c_s=s(s+1). Prove that (cm+1−ck)(cm+2−ck)⋯(cm+n−ck)(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k) is divisible by c1c2⋯cnc_1c_2\cdots c_n.
Step 3 of 6: Use the first block
Previous stepNext step
n!∣∏i=1n(m−k+i)n!\mid\prod_{i=1}^n(m-k+i)
Detailed analysis

The product of any nn consecutive integers is divisible by n!n! (equivalently, use the integral generalized binomial coefficient, with sign if the starting integer is negative).