Ok, so maybe I'm missing some crucial steps here, but I just can't see it.
So, I have got $\gcd(144,252) = 36$
I need to find ALL positive solutions. I can find $x=2$ and $y=-1$ by reversing the algorithm. The answer is given as follows.
$36=144-108 =144-(252-144) =2\times144-252 $
But then says
$4212=118 \times2 \times144 -117 \times252$
solution of $4212 = 144x + 252y$ are $x= 234- 252t/36 = 243 - 7t$ $ y= -117 + 144t/36 = -117 + 4t$
$x,y>0$ if $234/7 > t > 117/4$
FIRSTLY - where on earth does the multiplier $117$ come from? this seems to have been plucked from thin air,
and secondly, please, how to convert the $t$ values into the $(x,y)$ values. ie, how does $30$ relate to $(24,3)$