I had a problem with this question.
upon solving we get 21 * 21^1/2 = a * b^1/2

After this greg writes the values of a and b as 21 and 21. Can we do that? Or is there a shortcut to this?

And also aren’t we supposed to make “b” as small as possible?
Yes  this is just equating the two components essentially.
Can you make b any smaller while ensuring that a is an integer?
can you please show me how you equated it.
So we have
21 \times \sqrt{21} = a \times \sqrt{b}
Notice that when I put a = b = 21, I get the same thing as the LHS.