The conversation between human and AI assistant.
Does b!^k divide (b*k)! if b,k are positive integers?
Yes, please do.
Well, first let's expand (b*k)! as b*k * (b*k-1) * ... * (b+1)*b * (b-1)*(b-2) * ... * 1. Now, notice that b! appears as a factor in the product b*(b-1)*(b-2)*...*1, k times. So, we can rewrite (b*k)! as (b!)^k * (k!) * (b+k)*(b+k-1)*(b+k-2)*...*(b+1). Since (b!)^k is a factor of (b*k)!, it follows that it divides (b*k)!.