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 6 of 6: Identify the divisor
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c1c2⋯cn=n!(n+1)!c_1c_2\cdots c_n=n!(n+1)!
Detailed analysis

Finally, c1c2⋯cn=∏s=1ns(s+1)=n!(n+1)!c_1c_2\cdots c_n=\prod_{s=1}^n s(s+1)=n!(n+1)!. The two block divisibilities therefore prove c1c2⋯cn∣Pc_1c_2\cdots c_n\mid P.