Solution of the problem
"All cases of the maximum number of Alg/Euclidus steps,
	when 40<=a<=45, 1<=b<=8"
	   by case analysis

A trivial observation: #steps <= b

The tree of cases:

a,8	(no reminders 6,7)
	8,5	5,3	3,2	2,1 	- 5 steps
	8,4
	from ,3 - no chance to beat/repeat the record,
					   since adds maximum 3 steps

a,7	(no reminder 4)
	7,6	6,1
	7,5	5,2	2,1
	no chance
a,6
	6,5	5,1
	6,4	4,2
	no chance
a,5
	5,4	...
	no chance

no chance
-------------------
The answer: 5 steps