Given: ax + by=1
where a,b,x,y are integers.
which of following the statement is false.
i) (a.b)=1
ii) (x,y)=1
iii)(a,y)=1
iv) (b,y)=1
To determine which statement is false, we will substitute the given values into the equation and evaluate each statement:
Given: ax + by = 1
i) (a.b) = 1:
To evaluate this statement, we need to determine whether a and b are coprime (i.e., their greatest common divisor is 1). Since the equation does not explicitly mention the relationship between a and b, this statement does not provide enough information to determine its truth or falsehood. So, we cannot say whether statement i) is true or false.
ii) (x,y) = 1:
This statement suggests that x and y are coprime, i.e., their greatest common divisor is 1. It is unrelated to the equation in question, so we also cannot determine its truth or falsehood based on the given equation. Therefore, we cannot say whether statement ii) is true or false.
iii) (a,y) = 1:
In this statement, we consider the greatest common divisor of a and y. Since the equation does not provide enough information about the relationship between a and y, this statement is not guaranteed to be true or false. We cannot determine its truth or falsehood from the given equation. So, we cannot say whether statement iii) is true or false.
iv) (b,y) = 1:
Here, we evaluate the greatest common divisor between b and y. The equation does not explicitly mention b and y, so we cannot determine whether this statement is true or false based on the given equation. Therefore, we cannot say whether statement iv) is true or false.
In summary, we cannot determine the falsehood or truthfulness of any of the given statements (i), ii), iii), or iv) based solely on the equation ax + by = 1.