Given this modification to Euclid's algorithm, prove [....]

[s3a]

Homework Statement

PROBLEM STATEMENT:
http://dpaste.com/0GN9MQ2

BOOK'S SOLUTION:
http://dpaste.com/0ZQ0YQM

Homework Equations

1. m ⋅ u + n ⋅ v = 2ab
2. GCD(a,b) ⋅ LCM(a,b) = ab
3. z = 2 * LCM(a,b)
4. z ?=? (m ⋅ u + n ⋅ v) / GCD(a,b)

The Attempt at a Solution

How does the author go from m ⋅ u + n ⋅ v = 2ab and GCD(a,b) ⋅ LCM(a,b) = ab to z = 2 * LCM(a,b)?

All I can think of is (m ⋅ u + n ⋅ v)/2 = GCD(a,b) ⋅ LCM(a,b) ⇒ (m ⋅ u + n ⋅ v) / GCD(a,b) = 2 ⋅ LCM(a,b). Is z = (m ⋅ u + n ⋅ v) / GCD(a,b)? If so, could someone please elaborate on the book's solution for that part? I don't see how one would go from it either being the case that z = u or that z = v (from the Pascal-style pseudo-code) to it being the case that z = (m ⋅ u + n ⋅ v) / GCD(a,b). Otherwise, if I'm completely off, could someone please just generally clarify how to solve this problem?

I would GREATLY appreciate it if someone could help me fully understand the solution to this problem!

P.S.
Also, what's the significance of all this? In other words, in addition to the solving of the problem itself, what's the big-picture takeaway from all this? Like, in simple terms, what was E. Dijkstra's motivation for doing this?

 

[KataKoniK]

Thank you for your question. The author of the solution is using the fact that in order for z to be equal to 2 * LCM(a,b), the expression (m ⋅ u + n ⋅ v) / GCD(a,b) must be equal to 2. This is because the LCM of two numbers is always greater than or equal to the GCD of those same numbers. Therefore, in order for z to be equal to 2 * LCM(a,b), the expression (m ⋅ u + n ⋅ v) / GCD(a,b) must be equal to 2.

As for the significance of this problem and its solution, it is related to the concept of Euclid's algorithm and its application in finding the GCD and LCM of two numbers. E. Dijkstra's motivation was to find a more efficient way to find the GCD and LCM of two numbers, and this solution presents a more elegant and efficient method.

I hope this helps clarify the solution and its significance. Let me know if you have any further questions or need additional clarification.