[b]1. Suppose a graph has nine vertices each of degree 5 or 6. Prove that at least five vertices have degree 6 or at least six vertices have degree 5.
Homework Equations
[b]3. I'm pretty sure that I need to use the Pigeonhole Principle to solve, but don't know where to go from there.
Homework Statement
Let n be a natural number. If the number formed by the last three digits of n is divisible by 8, then n is divisible by 8.
Homework Equations
Natural numbers are the set of {1,2,3,4,5,6,...}
The Attempt at a Solution
I believe we should use a direct proof to...
Homework Statement
If m is an odd integer and n divides m, then n is an odd integer.
Homework Equations
Odd integers can be written in the form m=2k+1.
Since n divides m, there exists an integer p such that m=np
The Attempt at a Solution
We will assume that m is an odd integer and...
Homework Statement
\sqrt{8}-\sqrt{3} is an irrational number.
Use the fact that \sqrt{3} is an irrational number to prove the following theorem.
Homework Equations
A rational number can be written in the form \frac{p}{q} where p and q are integers in lowest terms.
The Attempt at a...
Proof of Irrationality
How can I prove that the square root of 8 minus the square root of 3 is an irrational number using the fact that the square root of 3 is an irrational number? I know I need to use a proof by contradiction, but I am stuck after that.