I found this problem while randomly scrolling through MathCounts problems to practice:

If x and y must be positive integers no greater than 100, how many ordered pairs (x, y) exist such that the least common multiple of x and y is ten times the greatest common factor of x and y?

Find the number of distinct ordered pairs (a,b) of positive integers a and b such that the LCM of a and b is 2024.

when multiply fractions with unlike fraction should I
multiply the numerator to numerator, denominator to denominator then simplified
or
find LCM of the denominator first then do the multiplication

Im a bit confused

Just wondering as to what problems (and therefore subjects I should focus on) would constitute the AMC 8:

Here's my guess:

Prealgebra/Logic - 3
Algebra - 8
Geometry - 5
Number Theory - 4
Probability/Counting - 5

(btw, you can elaborate on each section like

Number Theory - 4
Factorization - ?
LCM/GCD - ?
etc...

)

I know the Euclidean algorithm can be used to find the GCD of very large numbers. But, (unless you want to use the relationship between LCM and GCD), there isn't a simple algorithm (that I'm aware of) that will work for large LCMs.

Example

That example wasn't that bad. But some cases are even worse. Does an algorithm exist for this?

Suppose m and n are positive integers greater than 1. The product of their GCD and LCM is 35. Find m+n.

There's this question on my math homework that says "what are the factors of ?" (We're working on factors, divisors, LCM, GCD, and stuff like that in class.) I searched it up, and I'm getting two distinct answers: there are no factors of , or every integer is a factor of . I'm not sure which one is correct, if either one is correct at all. Does someone have the actual answer? thanks xD

Find the number of unordered pairs of natural numbers and such that .

Find the lcm of