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 1 of 6: Factor the difference
Previous stepNext step
ca−cb=(a−b)(a+b+1)c_a-c_b=(a-b)(a+b+1)
Detailed analysis

Since cs=s(s+1)c_s=s(s+1), direct expansion gives ca−cb=(a−b)(a+b+1)c_a-c_b=(a-b)(a+b+1).