The aim of this 200 page book is to enable talented students to tackle the sort of problems on number theory which are set in mathematics competitions. Topics include primes and divisibility, congruence arithmetic and the representation of real numbers by decimals. A useful summary of techniques and hints is included. This is a fully revised and extended edition of a book which was initially published as part of the composite volume 'Introductions to Number Theory and Inequalities'. The author was an Oxford University lecturer and teacher at Clifton College, Bristol. For ages 16+ competition preparation.