- #1

wubie

**Divisibility of "c" by "a "and "b" but not "ab".**

Hello,

I am having trouble with this question:

i) Give an example of three positive integers a,b,c such that a|c and b|c but ab does NOT divide c.

ii) In the situation of part (i), is there a condition that guarentees that if a|c and b|c, then ab|c?

iii) Is the condition in part (ii) necessary? Either prove that it is necessary, or give an example to show that it is not necessary.

It took me a long time to find three integers that satisfied conditions in part i) of the question. In fact, I couldn't find any such integers, someone had to tell me.

The integers that were given to me were 2,4,12. Now that these integers were given to me, I definitely can see how they satisfy conditions in part i).

However I still am having trouble with (ii) and obviously (iii).

For part i) I was playing around with prime numbers. And I couldn't find any integers which satisfied part i). Was that my mistake? Was that the reason I couldn't find the integers necessary to answer part i)?

As well, is that the key to ii) and iii)? That is, to guarantee that, if a|c and b|c, then ab|c, a and b must be prime numbers?

Any help would be appreciated. Thankyou.