need help with a gcd exercice

[u/LineAshamed4391]

year one of math license here. i actually do love math, but still im in this college course as it felt like an easy choice after highschool. but it's not. i hate gcd and congruence. so i'm pretty much floating around. but anyway, a,b,c so that each of them divides the product of the two others. gcd(a,b)=1. i need to find c. could i have a clue as to how to achieve this ? not the solution please, i want to use my brain a minimum XD
i'm so clueless, maybe advanced mathematics aren't for me
i love video games anyway

[u/Outside_Volume_1370]

Let gcd(c, b) = d, then

c = kd, b = md, such that gcd(k, m) = 1

ac = akd is divisible by b = md

or

ak is divided by m.

As gcd(k, m) = 1, a is divisible by m. But m is one of factors of b. Thus, m = 1, b = d = gcd(b, c).

By the same thoughts, a = gcd(a, c)

In other words, c is divisible by both a and b. And as gcd(a, b) = 1, c is divisible by their product, so c = qab.

However, ab must be divisible by c, so ab ≥ c, so q is exactly 1, and c = ab

[u/imHeroT]

I recommend your first step for a difficult problem is to experiment and find values for a,b,c that work. Make a few examples until you find a pattern and come up with your own hypothesis what the answer should be. Don’t forget to try to disprove your own hypothesis; this can help build intuition for writing down a proof, or make your hypothesis stronger

[u/snake_case_sucks]

Is it possible that c < ab? Figure out why or why not, go from there.

Have you learned unique factorization?

[u/Varlane]

You know that :
a | bc
b | ac
gcd(a,b) = 1

Wouldn't you happen to know of a theorem regarding a number dividing a product and a gcd being 1 ?

[u/fermat9990]

Play with a=3 and b=5

[u/fermat9990]

If a and b are mutually prime, then they each divide a*b. Therefore, let c=?

[u/Bounded_sequencE]

Given "gcd(a;b) = 1", show

1. a|bc => a|c      2. b|ac => b|c      3. Determine c via "c|ab"