Section 1.1

# Introduction

Number theory is all about integers, primarily their properties under multiplication and addition. The cornerstone of the multiplicative theory is *divisibility*, so this first section is focused on giving a solid foundation on this topic.

As with most things mathematical, we'll start with a definition.

Definition: An integer

ais said to divide another integerbif there exists an integercsuch thatb = ac. We write $a \mid b$ to mean “adividesb.”The integer 9 divides the integer 18 because $18 = 9 \cdot 2.$

The integer 2 does not divide the integer 5 because it is impossible to find an integercso that $5 = 2 \cdot c.$

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