Find the maximum and minimum value from integer division?

Question

Mathematics

Good day!
You are given three unknown integers x, y and z.
You know the integer division values ​​r = z / x and l = z / y.
What is the minimum and maximum possible division by integer y / x?
How to solve, tell me?

Answer 1

UPD. Rewrote from scratch.
r <= z/x < r + 1
l <= z/y < l + 1
Then y/x = z/x : z/y > r / (l+1)
On the other hand: y/x < z/ x : z/y < (r+1)/l
While I see the range [r / (l+1)]; [(r+1)/l]
The second is minus one if it's an exact integer: you can get closer, but you can't reach it. Simulate it like this…
[r / (l+1)]; [r/l]
Let's show that the bounds are reachable (for example, the first one). It touches on two inequalities.
z/x < r + 1
z/y >= l
By choosing x, y and z, we can bring the second to equality and approximate the first as much as we like.
UPD2. Let r = [z/x] = 10, l = [z/y] = 3
Then [y/x] will be between [10/4] = 2 and [10/3] = 3. parts - then from 2.5 to 3 2/3.
z=100.000, x=10.000, y=25.001, y/x=2.5001
z=100.000, x=9091, y=33.333, y/x ≈ 3.66659

Answer 2

If I understood the question correctly, then r and l are also integers.
Therefore, y=z/l and x=z/r.
As a result, y/x=(z/l)/(z/r)=r/l.

Answer 3

Here you need, as for me, to use the minimax function, and then you can solve it in the matlab mathematical package