Olympiad Number Theory: Challenging Problems

Olympiad Number Theory Through
Challenging Problems
Justin Stevens
THIRD EDITION

Contents
1 Divisibility 4
1.1 Euclidean and Division Algorithm . . . . . . . . . . . . . . . . . . 5
1.2 Bezout’sIdentity ........................... 16
1.3 Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . 22
1.4 Challenging Division Problems . . . . . . . . . . . . . . . . . . . . 27
1.5 Problems................................ 35
2 Modular Arithmetic 38
2.1 Inverses ................................ 38
2.2 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . 40
2.3 Euler’s Totient Theorem and Fermat’s Little Theorem . . . . . . . 45
2.4 The equation x2≡ −1 (mod p) ................... 57
2.5 Order ................................. 60
3 p-adic Valuation 69
3.1 Definition and Basic Theorems . . . . . . . . . . . . . . . . . . . . 69
3.2 p-adic Valuation of Factorials . . . . . . . . . . . . . . . . . . . . 72
3.3 Lifting the Exponent . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 General Problems for the Reader . . . . . . . . . . . . . . . . . . 84
4 Diophantine equations 85
4.1 Bounding ............................... 85
4.2 The Modular Contradiction Method . . . . . . . . . . . . . . . . . 85
4.3 General Problems for the Reader . . . . . . . . . . . . . . . . . . 92
5 Problem Solving Strategies 93
5.1 Chicken Mcnuggets anyone? . . . . . . . . . . . . . . . . . . . . . 93
5.2 VietaJumping ............................ 98
5.3 Wolstenholme’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 BonusProblems............................ 109
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For Cassie Stevens.
January 30th, 2000 to July 17th, 2013
“You never know how strong you are until being strong is the only choice you have.”

1
Divisibility
In this chapter, we will explore divisibility, the building block of number theory.
This chapter will introduce many important concepts that will be used throughout
the rest of the book. Divisibility is an extremely fundamental concept in number
theory, and has applications including puzzles, encrypting messages, computer se-
curity, and many algorithms. An example is checking whether Universal Product
Codes (UPC) or International Standard Book Number (ISBN) codes are legiti-
mate.
Figure 1.1: An example of a UPC code.
In order for the 12 digit UPC code above to be legitimate, we order the digits
x1, x2, x3,··· , x12. The expression
3x1+x2+ 3x3+x4+ 3x5+x6+ 3x7+x8+ 3x9+x10 + 3x11 +x12
then must be divisible by 10. We indeed verify that the above code gives
0×3+3×1+6×3+0×1+0×3+0×1+2×3+9×1+1×3+4×1+5×3+2×1 = 60,
which is divisible by 10. Therefore the above UPC code is valid.
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Justin Stevens 5
1.1 Euclidean and Division Algorithm
When the concept of division is first introduced in primary school, quotients and
remainders are used. We begin with a simple picture that should be familiar to
the reader, and we explore its relevance.
Figure 1.2: Division in primary school.
Source: CalculatorSoup
The process above used to divide 487 by 32 can be formalized through the
division algorithm.
Theorem 1.1.1 (Division Algorithm).For every integer pair a, b, there exists
distinct integer quotient and remainders, qand r, that satisfy
a=bq +r, 0≤r < b
Proof. We will prove that this is true for when aand bare positive. The other
cases when one or both of aand bare negative follow very similarly. There are
two parts in this proof:
•Proving that for every pair (a, b) we can find a corresponding quotient and
remainder.
•Proving that this quotient and remainder pair are unique.
For proving the existance of the quotient and remainder, given two integers a
and bwith varying q, consider the set
{a−bq with qan integer and a−bq ≥0}.