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 5 of 6: Cancel the prime safely
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(n+1)!∣B(n+1)!\mid B
Detailed analysis

The binomial coefficient is an integer, so (n+1)!(n+1)! divides pBpB. Since pp is prime and p>n+1p>n+1, gcd⁡(p,(n+1)!)=1\gcd(p,(n+1)!)=1; hence (n+1)!(n+1)! divides BB.