I think the toughest memorization task is learning the multiplication table,
I disagree. Learning all of the methods to calculate the antiderivative (indefinite integral) of a function required far more memorisation. Of course, there is a very good reason for this: it's hard (in the sense of being undecidable in the general case). Although, I do wonder why people learn the multiplication table up to 12 * 12, since the table up to 10 * 10 together with any basic multiplication algorithm (which ultimately boils down to convolution of two sequences) is sufficient.
and the most complex widely taught algorithm is long division.
I believe that's been removed from the UK syllabus now, except for the equivalent algorithm for polynomials. I am in favour of this, since the only application of numeric long division is to prove that rational numbers have terminating or recurring decimal expansions. Sincerely, Adam P. Goucher