Math question?

Question

If anyone could explain how the following is done, it would be greatly appreciated!

Suppose that gcd(a,b)=1 and suppose further that a divides c and that b divides c. Show that the product ab must divide c.

Answer 1

Well I won't do the whole thing, but here's a hint. If the gcd(a,b) = 1 then at least one of the two (a or b) is a prime number. Which also means that if you factored c, one of the factors would have to be that same prime number since it is divisible by that number. That should be enough to go on.

Answer 2

Since gcd(a,b) = 1, we know there are integers x and y such that ax + by = 1; multiplying through by c, we have

(1) acx + bcy = c.

Now a divides c and b divides c tells us there are integers A and B such that aA = c and bB = c. Substituting into (1) we have a(bB)x + b(aA)y = c. Since ab divides the left side, ab must divide the right side, so ab divides c.

(Notice it is not enough to say that every prime which divides a divides c; what we need is that whatever power of a prime divides one divides the other.)

Answer 3

We will show that any prime that divides a divides c and any prime that divides b divides c.
Now, since those primes that divide a do not divide b and those that divide b do not divide a, we will have shown that ab divides c.

By the given that a divides c, it is clear that any prime p dividing a will divide c.
The same applies to b.

Thus, ab divides c.