Some Discrete Math Help, Im Exhausted

[thecleric]

Homework Statement

Suppose that we play the following game. You are given a pile of N matches. You break the pile into two smaller piles of m and n matches. Then you form the product 2mn and remember it. Next, you take one of the piles and break it into two smaller piles (if possible), say of m’ and n’ matches. You form the product 2m’n’ and add it to the 2mn that you had before, so now you have 2mn+2m’n’. You proceed again by breaking one of the piles into two and adding the resulting product. The process is finished when you finally have N piles of one match in each. By convention, if N = 1 then you don't do anything and the result is 0. Try to take a pile of five matches and play this game several times, each time breaking to piles in a different way. What do you see?

If you start with a pile of matches, no matter how you break it, the sum of the computed products will always be .

Homework Equations

The Attempt at a Solution

Can someone explain this to me?

Homework Statement

Prove that A_n<$$\left($$7/4)ⁿ

Homework Equations

n greater than or equal to 3

The Attempt at a Solution

Homework Statement

Prove statement below by contrapositive and contradiction:
If a prime number divides the square of an integer, then that prime number divides that integer.

Homework Equations

n is prime for all positive integers r and s if n=rs where r=1 or s=1

The Attempt at a Solution

Basically let m be a prime number, if m divides n² then m divides n

 

[thecleric]

anyone?

 

[matt grime]

For the first one you should just do some examples, as it suggests.

For the second. You didn't define A_n.

For the third, it is not "Basically if m divides n^2 then m divides n", since that is clearly false: 4 divides 4, and 4 does not divide 2. Of course 4 isn't prime. So what do you know about primes?

 

[thecleric]

sorry for the 2nd one its for n is greater than or equal to 3.

n is prime for all positive integers r and s if n=rs where r=1 or s=1

**fixed the original post

 

[matt grime]

No, you have still not defined A_n. Telling me that n must be at least 3 doesn't help.

Do know of any other ways to define primality? Such as p is prime if p divides ab implies p divides a or p divides b? Can you prove that this definition is equivalent to yours? ANd can you see how it helps?