ch1.1.4-integers-and-factors

Chapter IV – Of the Nature of Whole Numbers with respect to their Factors

Summary: Distinguishes prime numbers (no non-trivial factors) from composite numbers (products of smaller factors), and shows that every composite number decomposes into prime factors.

Sources: chapter-1.1.4

Last updated: 2026-04-24

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§37–38: Products and Factors

A product arises from multiplying two or more numbers; those numbers are its factors. For instance, in $abcd$, the letters $a,b,c,d$ are the four factors (source: chapter-1.1.4).

§39–40: Prime Numbers

Numbers that cannot be represented as a product of two smaller numbers (other than $1\times n$) are called simple or prime numbers. Unity (1) is not counted as a factor for this purpose.

The sequence of primes begins: $2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,\dots$

Euler notes that no one has discovered a regular law governing the gaps between primes (source: chapter-1.1.4).

§41: Composite Numbers

All other numbers are composite: they can be factored into primes. Any non-prime factor can itself be broken down, so every composite number ultimately equals a product of prime numbers alone. Example: $30=5\times 6=5\times 2\times 3$ (source: chapter-1.1.4).

§42–43: Prime Factorisation

To find all prime factors, divide repeatedly by the smallest prime that divides the number:

$360=2\times 180=2^2\times 90=2^2\times 2\times 45=2^3\times 3\times 15=2^3\times 3^2\times 5$

(source: chapter-1.1.4)

§44: Connection to Division

To find prime factors in practice, division is required; this motivates the next chapter on division.

Related pages

• prime-numbers
• ch1.1.3-multiplication-simple-quantities
• ch1.1.5-division-simple-quantities
• ch1.1.6-properties-integers-divisors