**PROBLEMS**

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,

2.) **Solution:**

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,

Hence, then