Here's question I'm really struggling with:
So far I believe I have found $d=21$ and $x=-2$ and $y=5$.
From here I'm unsure where to go as part b is making very little sense, could someone explain a way to go about it?
I have used standard trial and error to find more pairs of integers that satisfy the identity as it asked in b(i), however I wonder if there is a mathematical solution.
What you did for part a is correct.
Now, part b is just to prove that the numbers $-2+105k$ and $5-252k$ are also solutions, for each $k\in\mathbb N$. And whoever created the problem asked you to prove this assertion by induction (a silly idea, in my humble opinion).