I'm given: $4x+51y=9$.
I am given a hint that when we use $4x=9 \pmod{51}$ we get $x = 15 + 15t$, and also if we use the congruence $51y=9 \pmod 4$ we get $y=3+4s$. They say it's handy to then find the relation between $s$ and $t$.
I have no idea how they got those suggestions and I need to know how to do that too.
I'm really stuck :(
On
4x + 51y = 9 and 4x = 9(mod 51) then x = 15(mod 51) or 15 + 51t.
Substitute x = 15 + 51t into the original equation 4(15 + 51t) + 51y = 9.
60 + 204t + 51y = 9 then y = -1 - 4t. The complete solution is x = 15 + 51t and y = -1 - 4t.
On
[For the following, please refer to the figure that follows the essay (the figure is also available in PDF).]
The manipulations performed from steps (0) to (12) were designed to create the linear system of equations (0a), (7a) and (12a). The manipulations end when the absolute value of a coefficient of the latest equation added is 1 (see (12a)). It is possible to infer equation (0a), (7a) and (12a) from (0), (7) and (12) respectively without performing manipulations (0) to (12) directly. In every case, select the smallest absolute value coefficient, generate the next equation by replacing every coefficient with the remainder of the coefficient divided by the selected coefficient (smallest absolute value coefficient) – do the same with the right-hand constant – and add the new variable whose coefficient is the smallest absolute value coefficient. If the new equation has a greatest common divisor greater than one, divide the equation by the greatest common divisor. Stop when the absolute value of a coefficient of the latest equation added is 1.
Write: $4x + 51y = 9$ as $9x - 5x + 45y + 6y = 9$ $\to$ $9| (-5x + 6y)$ $\to$ $-5x + 6y = 9h$ $\to$
$-5x = 3(3h - 2y)$ $\to$ $3|(-5x)$ $\to$ $3|x$. $x = 3k$. Back to the main
equation: $4(3k) + 51y = 9$ $\to$ $51y = 9 - 12k$ $\to$ $17y = 3 - 4k$ $\to$ $y = \dfrac{3 - 4k}{17}$. In order
for $y$ to be an integer we must have: $k = 5 + 17t$. Thus $x = 3k = 3(5 + 17t) = 15 + 51t$. So
$y = \dfrac{3 - 4(5 + 17t)}{17} = \dfrac{-17 - 68t}{17} = -1 - 4t$.
Thus the solution is: $(x,y) = \{(15 + 51t, -1 - 4t): t \in \mathbb{Z}\}$.
Check: $4x + 51y = 4(15 + 51t) + 51(-1 - 4t) = 60 + 204t - 51 - 204t = 9$.