1.) Find a linear function such that and .
2.) Solve for :
3.) Prove that the product of consecutive numbers is always divisible by .
4.) Prove that if is prime, and are integers, and , then .
SOLUTIONS AND PROOFS
Post Date: October 20, 2009
1. Solution: This is just the same as saying, find the equation of the line passing through and . So, by point slope formula, we have,
3.) Proof: A number is divisible by if it is divisible by and . A product of consecutive numbers is divisible by because at least one of them is even, so it remains to show it is divisible by .
If a number is divided by , its possible remainders are and . Assume and be the three consecutive numbers, and be the remainder if is divided by .
Case 1: If , we are done.
Case 2: If , then
Case 3: If , then .
Since the product of the three consecutive numbers is even, and for each case of , one of the consecutive numbers is divisible by , the product of three consecutive numbers is divisible by
4.) Proof: From definition, for some
Raising both sides of the equation to , we have By the binomial theorem, .
Notice that every term aside from is divisible by . (Why?). Therefore,