integer division help

[shelly89]

a b c and d are integers with

\(\displaystyle a \not = 0 \)

\(\displaystyle b \not = 0 \)


Prove that if a/c and b / d then ab / cd.

am really unsure on how to go about this, I know that if a/c than we \(\displaystyle c = aq \) for some integer q

and \(\displaystyle d = bq \) for some integer q

so

\(\displaystyle c-d = aq-bq \)

\(\displaystyle c-d=(a-b)q \)


dont really know where to go from here, any help appreciated!

 

[tkhunny]

1) Can you multiply the fractions?
2) You have d = qb, where did you get c = qa? Why would 'q' appear in both?
3) You're looking for cd, why did you wander into c-d?

 

[shelly89]

Okay, thank you, I was trying to follow my notes, and copy the example in my book, i dont actually know what I was doing....


but as you say multiply the fractions,


well than

\(\displaystyle \frac{a}{c} \times \frac{b}{d} = \frac{ab}{cb} \)

I knw i have to use the result that given any 'a' and 'b' there exist unique integers q and r such that

a = qb+r


am I on the 'right' path?


so is it cb= qab ?

 

[JeffM]

a b c and d are integers with

\(\displaystyle a \not = 0 \)

\(\displaystyle b \not = 0 \)


Prove that if a/c and b / d ARE WHAT then ab / cd IS WHAT. This question is incomplete. We are not mind readers. Verbs and predicates are needed to make comprehensible sentences.

am really unsure on how to go about this, I know that if a/c than we \(\displaystyle c = aq \) for some integer q

and \(\displaystyle d = bq \) for some integer q

so

\(\displaystyle c-d = aq-bq \)

\(\displaystyle c-d=(a-b)q \)


dont really know where to go from here, any help appreciated!

Please give the question exactly as it is given in your book. We sincerely appreciate that you have told us your thoughts about the question, but we cannot evaluate those thoughts and make helpful suggestions if we are not completely sure of what you are supposed to be thinking about.

 

[Mrspi]

a b c and d are integers with

\(\displaystyle a \not = 0 \)

\(\displaystyle b \not = 0 \)


Prove that if a/c and b / d then ab / cd.

am really unsure on how to go about this, I know that if a/c than we \(\displaystyle c = aq \) for some integer q

and \(\displaystyle d = bq \) for some integer q

so

\(\displaystyle c-d = aq-bq \)

\(\displaystyle c-d=(a-b)q \)


dont really know where to go from here, any help appreciated!

I think it would clear up much confusion if you had indicated what you MEAN by a/c and b/d. It is clear to me, based on your attempt at a proof, that you intend this to mean "a DIVIDES c" and "b DIVIDES d".

If a and c are integers, and a is not equal to 0, a divides c means that there exists some integer (let's call it q) for which c = a*q.

Similarly, if b and d are integers and b is not equal to 0, b divides d means that there exists some integer (let's call it r) for which d = b*r.

Then, if we are considering whether or not (ab) divides (cd), we can simply substitute the expressions we've found above for c and d, namely.....

cd = aq*br
or, using the commutative and associative properties of multiplication, we can rewrite the right-hand side as

cd = (a*b)*(q*r)

Because the integers are closed for multiplication, c*d, a*b, and q*r are all integers. Because neither a nor b is equal to 0, a*b is not equal to 0 either.

We can express the integer c*d as the product of a non-zero integer a*b and some other integer q*r. This, by definition, means that a*b divides c*d.

I think that should do it.......