ch1.1.6-properties-integers-divisors

Chapter VI – Of the Properties of Integers with respect to their Divisors

Summary: Classifies integers into residue classes for each divisor, shows how all divisors of a number are derived from its prime factorisation, and includes a table of all divisors for integers 1–20.

Sources: chapter-1.1.6

Last updated: 2026-04-24

---

§58–62: Residue Classes

For any divisor $n$, every integer falls into one of $n$ residue classes according to its remainder:

Divisor	Classes
2	$2a$ (even), $2a+1$ (odd)
3	$3a$, $3a+1$, $3a+2$
4	$4a$, $4a+1$, $4a+2$, $4a+3$
5	$5a$, $5a+1$, $5a+2$, $5a+3$, $5a+4$

Even numbers take the form $2a$; odd numbers take the form $2a+1$ (source: chapter-1.1.6).

§63–65: Divisors from Prime Factorisation

If $n=p\cdot q\cdot r\cdots$ (prime factors), then $n$ is divisible by every product formed from subsets of those primes. For $60=2\times 2\times 3\times 5$, the divisors include $1,2,3,4,5,6,10,12,15,20,30,60$. The complete list of divisors is obtained by taking the prime factors one-by-one, two-by-two, three-by-three, etc. (source: chapter-1.1.6).

§66: Prime Numbers Revisited

Every number is divisible by 1 and by itself. A number with no other divisors is prime. All others (composite numbers) have additional divisors (source: chapter-1.1.6).

Table of Divisors (1–20)

$n$	Divisors	Count	Prime?
1	1	1
2	1, 2	2	P
3	1, 3	2	P
4	1, 2, 4	3
5	1, 5	2	P
6	1, 2, 3, 6	4
7	1, 7	2	P
8	1, 2, 4, 8	4
9	1, 3, 9	3
10	1, 2, 5, 10	4
11	1, 11	2	P
12	1, 2, 3, 4, 6, 12	6
13	1, 13	2	P
14	1, 2, 7, 14	4
15	1, 3, 5, 15	4
16	1, 2, 4, 8, 16	5
17	1, 17	2	P
18	1, 2, 3, 6, 9, 18	6
19	1, 19	2	P
20	1, 2, 4, 5, 10, 20	6

(source: chapter-1.1.6)

§67: Zero

Zero is divisible by every number: $0÷a=0$ for all $a$, since $a\times 0=0$ (source: chapter-1.1.6).

Related pages

• prime-numbers
• ch1.1.4-integers-and-factors
• ch1.1.5-division-simple-quantities
• ch1.1.7-fractions-in-general