## 104 Number Theory Problems: From the Training of the USA IMO TeamThis book contains 104 of the best problems used in the training and testing of the U. S. International Mathematical Olympiad (IMO) team. It is not a collection of very dif?cult, and impenetrable questions. Rather, the book gradually builds students’ number-theoretic skills and techniques. The ?rst chapter provides a comprehensive introduction to number theory and its mathematical structures. This chapter can serve as a textbook for a short course in number theory. This work aims to broaden students’ view of mathematics and better prepare them for possible participation in various mathematical competitions. It provides in-depth enrichment in important areas of number theory by reorganizing and enhancing students’ problem-solving tactics and strategies. The book further stimulates s- dents’ interest for the future study of mathematics. In the United States of America, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), the American Invitational Mathematics - amination (AIME), and the United States of America Mathematical Olympiad (USAMO). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately ?fty very promising students who have risen to the top in the American Mathematics Competitions. |

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### Contents

1 | |

Division Algorithm | 4 |

Primes | 5 |

The Fundamental Theorem of Arithmetic | 7 |

GCD | 11 |

Euclidean Algorithm | 12 |

Bezouts Identity | 13 |

LCM | 16 |

Numerical Systems | 40 |

Divisibility Criteria in the Decimal System | 46 |

Floor Function | 52 |

Legendres Function | 65 |

Fermat Numbers | 70 |

Mersenne Numbers | 71 |

Perfect Numbers | 72 |

Introductory Problems | 75 |

The Number of Divisors | 17 |

The Sum of Divisors | 18 |

Modular Arithmetic | 19 |

Residue Classes | 24 |

Fermats Little Theorem and Eulers Theorem | 27 |

Eulers Totient Function | 33 |

Multiplicative Function | 36 |

Linear Diophantine Equations | 38 |

### Common terms and phrases

American Mathematics Competitions Andreescu answer arithmetic progression assume claim conclusion follows congruent consecutive integers consider contradiction cube denote digit sum distinct primes divides equal Euclidean algorithm Euler’s theorem exactly Example exists Feng Fermat number Fermat’s little theorem Find finite gcd(a gcd(a,b gcd(k gcd(m gcd(m,n gcd(n greatest common divisor Hence implying induction hypothesis inequality infinitely many primes integer greater integer n International Mathematical Olympiads introductory problem lcm(m least common multiple Mathematical Association mod 9 modular arithmetic nonnegative integers nonzero Note number theory obtain odd integer odd numbers odd positive integers pairwise relatively prime perfect square pigeonhole principle positive divisors positive integer less prime divisor prime factorization prime numbers Proposition 1.30 Prove real numbers residue classes modulo satisfies the conditions Second Proof sequence set of residue smallest Solution subset suffices to show unique USAMO Wilson's theorem wobbly number write