Linear Diophantine equation $3x + 5y = 11$

Question

Solve the Diophantine equation $3x + 5y = 11$

I know how to calculate GCD

$$5 = 1\cdot 3 + 2$$

$$3 = 1\cdot 2 + 1$$

$$2 = 2\cdot 1 + 0$$

But how do I use this theorem to derive the correct answer?

Answer 1

Hint $\ $ By inspection/Euclid $\,\ 3\cdot 2 - 5\cdot 1\, =\, 1.\ $ Scale that by $11$

Remark $\ $ See here for a convenient version of the Extended Euclidean Algorithm, which also explains how to obtain the general solution from a particular solution.

Answer 2

3x + 5y = 11 is equivalent to 3x = 11(mod 5). Solving x = 7(mod 5) and x = 7 + 5k. Substituting x = 7 + 5k into the original equation, you get y = -2 - 3k. The complete solution is x = 7 + 5k and y = -2 - 3k.

Answer 3

$$3x+5y=11(3\cdot2-5\cdot1)\iff3(x-22)=-5(11+y)$$

$\displaystyle\implies\frac{3(x-22)}5=-11-y$ which is an integer

$\displaystyle\implies5|3(x-22)\iff 5|(x-22)\iff x\equiv22\pmod5\equiv2$

Answer 4

What you found is: $$\begin{align}5 &= 1\times3 + 2,\\ 3 &= 1\times2 + 1.\end{align}$$

So we have: $$\begin{align}1 &=3-1\times2 \\&=3-1\times(5-1\times3) \\&=3\times2-5.\end{align}$$

Now we have a particular solution of $3x+5y=1$: $(2,-1)$.

Use this to get a particular solution of $3x+5y=11$: $(22,-11)$, and with this, find a general solution of the equation. You might know how to conclude.

Answer 5

Alternatively, you've already found the GCD of 3 and 5 using the Euclidean Algorithm.

$\gcd(3,5)=1$

Since the GCD divides both sides of the equation in that $1|3x$ and $1|5y$, and $1|11$ we know that the diophantine equation has a solution.

A diophantine equation in the form $ax + by = c$ can also be expressed as two congruences; the first $\pmod a$ and the second $\pmod b$. So the equation $3x +5y = 11$ can also be expressed as two congruences:

$5y \equiv 2 \pmod 3 $ and $3x \equiv 1 \pmod 5$

After finding a particular solution, $x_0$ and $y_0$, we can find all values of $x$ and $y$ by remembering that:

$x=x_0 + 5t$ and $y=y_0 - 3t$ where $t$ is an integer.

Or in general, $x=x_0 + bt$ and $y=y_0 - at$ where $t$ is an integer.

One particular solution is $x_0=2$ and $y_0=1$. The other solutions can easily be found.