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Number Theory - Art of Problem Solving

Number TheoryNaoki PrefaceThis set of notes on Number Theory was originally written in 1995 for studentsat the IMO level. It covers the basic background material that an IMOstudent should be familiar with. This text is meant to be a reference, andnota replacement but rather a supplement to a Number Theory textbook;several are given at the back. Proofs are given when appropriate, or whenthey illustrate some insight or important idea. The problems are culled fromvarious sources, many from actual contests and olympiads, and in generalare very difficult. The author welcomes any corrections or DivisibilityFor integersaandb, we say thatadividesb, or thatais adivisor(orfactor) ofb, or thatbis amultipleofa, if there exists an integercsuchthatb=ca, and we denote this bya|b. Otherwise,adoes not divideb, andwe denote this bya-b. A positive integerpis aprimeif the only divisors ofpare 1 andp.

The Division Algorithm. For any positive integer a and integer b, there exist unique integers q and r such that b = qa + r and 0 ≤ r < a, with r = 0 iff a | b. 1. ... If a polynomial with integer coefficients factors into two polynomials with rational coefficients, then it factors into two poly-nomials with integer coefficients.

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