How to prove this simple number theory problem If the product of two numbers is an integer in the form of 3K-1 or 4k-1 or 6k-1, then there must be an integer in the form of 3K-1 or 4k-1 or 6k-1?

How to prove this simple number theory problem If the product of two numbers is an integer in the form of 3K-1 or 4k-1 or 6k-1, then there must be an integer in the form of 3K-1 or 4k-1 or 6k-1?

Let these two numbers be x and y,
If XY ≡ 1 ≡ 2 (MOD3)
Then x ≡ 1 (MOD3), y ≡ 2 (MOD3) or X ≡ 2 (MOD3), y ≡ 1 (MOD3)
If XY ≡ 1 ≡ 3 (mod4)
Then x ≡ 1 (mod4), y ≡ 3 (mod4) or X ≡ 3 (mod4), y ≡ 1 (mod4)
If XY ≡ 1 ≡ 5 (mod6)
Then x ≡ 1 (mod6), y ≡ 5 (mod6) or X ≡ 5 (mod6), y ≡ 1 (mod6)
So the proposition holds

A proof of number points
The function f (x, y) is shown in the figure
It is proved that the function f (x, y) is continuous at the origin, and the directional derivative along any direction exists, but it is not differentiable

As shown in the picture