Some New Theorems on Divisibility

Extract

Introduction. If we call the number pa + qb the T_p,q transform of the number n = 10a + b, then in general the T_p,q transform of n will differ from n; it easily seen, however, that given any number n, we can always find a pair p, such that the T_p.q transform of n is n itself. In fact, if n = 10a + b and p = 10 − b q = a + 1 then the T_p,q transform of n is (10 − b) a + (a + 1)b = 10a + b = n.