The discussion revolves around finding pairs of integers \(a\) and \(b\) such that their greatest common divisor (GCD) is 14 and their least common multiple (LCM) is 168. Participants explore the relationship between GCD and LCM through the equation \(a \times b = \text{gcd}(a,b) \times \text{lcm}(a,b)\).
The conversation is ongoing, with participants sharing insights and questioning each other's reasoning. Some have suggested methods to streamline the search for pairs, while others are still grappling with the concepts involved. There is no explicit consensus on the number of valid pairs, and participants are exploring various approaches.
Participants mention the challenge of finding all pairs without resorting to exhaustive trial and error. There is a concern about time constraints related to upcoming tests, which adds pressure to find a quicker solution.
Maybe, maybe not. What does it tell you?a x b = gcd(a,b) x lcm(a,b) (useful?)
That looks correct. Is it useful?bphysics said:a x b = 2352...
You don't know how to find all of the solutions to a x b = 2352?bphysics said:it's somewhat useful, gives me one more piece of information, but i can't see the way to solve it...
Ah. So you do know how to solve the problem, you're just fishing for other methods.bphysics said:i can find pairs, but some of them would be incorrect ex. a = 1 and b = 2352. i was wondering if there would be a better way or shortcut without having to go through all the pairs.
Er, what is there to explain? You know how to find all the solutions to a x b = 2352, so now we're just imposing some of the information from the original problem to streamline our work.bphysics said:wait a dividing lcm(a, b)?
can you explain more?
I thought that's what I was doing...bphysics said:could you lead me though it?
Hrm. I expected three answers. (But I could be wrong)bphysics said:so are the answers (14, 168) and (42, 56)?