If ax + by = 1,where a, b, x and y are intege | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The equation $ax + by = 1$ is a linear Diophantine equation.According to Bezout's identity,the equation $ax + by = c$ has integer solutions for $x

Vedclass · Accepted Answer

$(b, y) = 1$

Vedclass · Answer

(C) The equation $ax + by = 1$ is a linear Diophantine equation.According to Bezout's identity,the equation $ax + by = c$ has integer solutions for $x$ and $y$ if and only if $\gcd(a, b)$ divides $c$.Here,$c = 1$,so $\gcd(a, b)$ must divide $1$,which implies $\gcd(a, b) = 1$. Thus,option $(d)$ is true.Similarly,for $ax + by = 1$,we can rewrite it as $ax = 1 - by$,which implies $\gcd(a, b) = 1$ and $\gcd(a, y) = 1$ is not necessarily true,but $\gcd(a, b) = 1$ is a necessary condition.Specifically,$\gcd(a, b) = 1$,$\gcd(a, x)$ is not necessarily $1$,$\gcd(b, x)$ is not necessarily $1$,and $\gcd(x, y)$ is not necessarily $1$.However,looking at the options,$\gcd(b, y) = 1$ is not a required condition for the existence of solutions to $ax + by = 1$. For example,if $a=1, b=2, x=1, y=0$,then $1(1) + 2(0) = 1$. Here $\gcd(b, y) = \gcd(2, 0) = 2 
eq 1$. Thus,option $(c)$ is not necessarily true.

If $ax + by = 1$,where $a, b, x$ and $y$ are integers,then which one of the following is not true?

Similar Questions

If $\alpha, \beta$ are the roots of $x^2-a(x-1)+b=0$, then the value of $\frac{1}{\alpha^2-a \alpha}+\frac{1}{\beta^2-a \beta}+\frac{2}{a+b}$ is:

If $a \in R$ and the equation $-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$ (where $[x]$ denotes the greatest integer $\leq x$) has no integral solution,then all possible values of $a$ lie in the interval

If the extreme value of $3x - 2x^2 + 1$ is $k$, then the set of all real values of $x$ for which $kx^2 + 2x + 1 > 0$ is

Let $\lambda \in R$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is $..........$

The set of all real values $a$ for which $-1 < \frac{2 x^2+a x+2}{x^2+x+1} < 3$ holds for all real values of $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module