The exercises in Section 1.2 were concerned with divisibility properties. In many cases rules were studied for a given divisor that gave a new number significantly smaller than the dividend, both having the same remainder upon division by the divisor (e.g., for divisibility by 2, 3, 4, 5, 8, 9, 10, and 11). In general, this relation that two numbers have the same remainder upon division by a given divisor proves to be such a useful tool in number theory that Gauss introduced a special notation to represent it that quickly became part of almost every mathematician’s working vocabulary. Taking the names from Latin, we call two such numbers congruent, and their divisor the modulus.
- 3.For the origins of these types of questions, see P. Erdős: Summa Brasiliensis Math. II (1950), pp. 113–123.Google Scholar
- 7.Lifsic’s proof appears as the solution to Problem 143, p. 43 and pp. 159–161 in N. B. Vasil’ev, A. A. Egorov: Zadachi vsesoyuznikh matematicheskikh Olimpiad (Problems of the All-Soviet-Union Mathematical Olympiads), Nauka, Moscow, 1988 (in Russian).Google Scholar
- 10.Remark of I. Z. Ruzsa, personal communication.Google Scholar