Number Theory Basics

Divisibility, prime factorization, GCD/LCM, and modular arithmetic — the study of integers that underlies cryptography and abstract algebra.

Prime factorization breaks a composite number into prime building blocks

Divisors of 60

123456101215203060

Prime factorization explains every divisor: choose some of the prime factors and multiply them.

Definition

Number theory studies the properties of integers — especially divisibility: when does one integer divide another evenly, with no remainder?

We say $a$ divides $b$ (written $a \mid b$ ) if there's an integer $k$ with $b = ak$ . A prime number is an integer greater than 1 whose only positive divisors are 1 and itself (2, 3, 5, 7, 11, …). Every other integer greater than 1 is composite — it can be built by multiplying smaller primes together.

Prime factorization

$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$ .

This is the unique way (up to order) to write 60 as a product of primes — a fact important enough to have a name: the Fundamental Theorem of Arithmetic.

Try it

Is 1 a prime number? Why does the standard definition exclude it?

Solution

No — 1 is excluded by definition. If 1 were allowed to be prime, prime factorizations would no longer be unique (e.g. $6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3 = \cdots$ ), breaking the Fundamental Theorem of Arithmetic. Excluding 1 keeps factorizations clean and unique.

Related concepts

Needs first

Order of Operations

Algebraic Properties Expressions

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