Proofs (2 Questions)

[bigmac4ever16]

---Show that if an integer m divides an integer n and p is an integer, then m divides (n)(p).






----Show that if an integer m divides an integer n and an integer p divides an integer q, then (m)(p) divides (n)(q).

 

[mmm4444bot]

Hi Mac:

Have you learned about the concept known as "closure" ?

For example, if we know that Q and P are both integers, then we also know that their product QP is also an integer because the set of integers is closed under the operation of multiplication.

(This is a formal way of saying, "The product of two integers is always an integer.")

Here's how we could use closure, in your first proof.

Given: M, N, and P are integers

Given: M divides N

Since M divides N, we know that N/M is an integer. Let's call this quotient Q.

N/M = Q

Now, we want to show that NP/M is an integer (i.e., M divides NP).

Factor out N/M.

(N/M)P

Replace N/M with Q.

QP

In other words, the division of NP by M equals QP:

NP/M = QP

But QP is a product of two integers, so it is an integer (due to closure).

Therefore, NP must be divisible by M.

Try using the same line of reasoning, on your second exercise.

Please show whatever work you can, if you would like more help.

Cheers ~ Mark