Sometimes nǃ is very divisible by n. For example, 12ǃ is divisible by 125 while 720ǃ is divisible by 720178. On the other hand, if p is prime then pǃ is divisible by p but not by p2.

In general it is quite difficult to calculate the maximum exponent m(n) such that nm(n) divides nǃ

In this short article I will locate some values of n, for which nǃ is very divisible by n, find a tight upper bound for m(n), and point the reader in the direction of related problems.

The best separation of substances by adsorption chromatography requires a de-sorbing solvent whose concentration-volume graph is concave. Various methods have been devised to achieve this, sometimes ignoring the density difference between the dilute and concentrated solvents. A particular method of mathematical interest is that of Baker, Ellis and Hawley [1] where standard components can be used and density differences are reduced by using a floating siphon.

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality

Sir Arthur Eddington died on 22nd November, 1944. The last sixteen years of his life had been occupied chiefly with the discovery and development of certain new principles in physics, which he used in the first place in order to calculate theoretically the “ Constants of Nature ” (such as the ratio of the mass of the proton to the mass of the electron) ; he did in fact succeed in deriving correctly the values of all those constants of Nature which are pure numbers. Later he showed that his ideas could be presented as a coherent system, to which he gave the name Fundamental Theory, and which may be broadly summed up in the statement: the external world is isomorphous with the square of quaternion algebra.

A-level courses and examinations are with us for some time to come. Even if Mevel is introduced, for most pupils who spend two years in the I sixth form A-levels will remain the basis of their studies. It seems to me that we need a major debate about A-level mathematics; we need to review it as it is and try to decide what developments are necessary. The discussion I paper ‘Mathematics at A-level’ has already raised a number of points, some of which are touched on here. This article is not a statement of official SMP policy; any opinions expressed in it are my own. The aim is to provoke discussion. I shall consider a number of issues which seem to me to be important, hoping that others will be stimulated to contribute comments and suggest other topics which need attention.