Problem Set 2
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