This is a pure Math problem. We need the knowledge of number theory to cover the proof and solution. No idea why microsoft uses this problem in real interview.

The basic idea is to use the property of Bézout's identity and check if z is a multiple of GCD(x, y)

Quote from wiki:

Bézout's identity (also called Bézout's lemma) is a theorem in the elementary theory of numbers:

let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x

and y such that ax+by=dIn addition, the greatest common divisor d is the smallest positive integer that can be written as ax + by

every integer of the form ax + by is a multiple of the greatest common divisor d.

If a or b is negative this means we are emptying a jug of x or y gallons respectively.

Similarly if a or b is positive this means we are filling a jug of x or y gallons respectively.

x = 4, y = 6, z = 8.

GCD(4, 6) = 2

8 is multiple of 2

so this input is valid and we have:

-1 * 4 + 6 * 2 = 8

In this case, there is a solution obtained by filling the 6 gallon jug twice and emptying the 4 gallon jug once. (Solution. Fill the 6 gallon jug and empty 4 gallons to the 4 gallon jug. Empty the 4 gallon jug. Now empty the remaining two gallons from the 6 gallon jug to the 4 gallon jug. Next refill the 6 gallon jug. This gives 8 gallons in the end)

See wiki:

and comments in the code

```
public boolean canMeasureWater(int x, int y, int z) {
//limit brought by the statement that water is finallly in one or both buckets
if(x + y < z) return false;
//case x or y is zero
if( x == z || y == z || x + y == z ) return true;
//get GCD, then we can use the property of Bézout's identity
return z%GCD(x, y) == 0;
}
public int GCD(int a, int b){
while(b != 0 ){
int temp = b;
b = a%b;
a = temp;
}
return a;
}
```