[Introduction to Number Theory Series Part 2] What Do Divisors and Multiples Show Us?

The first real relation in number theory is divisibility. Before we talk about gcds, primes, or congruence, we need to know what it means for one integer to divide another.

In this post we will work with positive divisors unless stated otherwise.

What this post covers

• What divisibility means
• How divisors and multiples express the same relation from opposite directions
• How quotient and remainder fit into the picture
• Why this language becomes the base of later topics

Key terms

• divisibility: the relation that says one integer divides another with remainder 0
• divisor: a number that divides another integer exactly
• multiple: a number obtained by multiplying another integer by a given integer
• quotient: the whole-number part in a division statement
• remainder: the value left over after division

What does divisibility mean?

For integers a and b, we write

$$a \mid b$$

when a divides b. This means there is some integer k such that

$$b = ak.$$

So a | b is a relation, not a quotient. It does not mean b/a written differently.

For example,

$$4 \mid 20$$

because

$$20 = 4 \times 5 + 0.$$

But 6 does not divide 20, because

$$20 = 6 \times 3 + 2,$$

and the remainder is not 0.

Divisors and multiples are two views of the same fact

If 3 | 12, then:

• 3 is a divisor of 12
• 12 is a multiple of 3

So divisor and multiple are not two unrelated ideas. They are the same divisibility relation viewed from different sides: the dividing side and the resulting side.

Why do quotient and remainder matter?

Once we try to divide one integer by another, we usually get two pieces of information:

• the quotient, which tells us how many whole copies fit
• the remainder, which tells us what is left over

For example,

$$18 = 5 \times 3 + 3.$$

Here the quotient is 3 and the remainder is 3. In other words, three whole copies of 5 fit into 18, and 3 is left over.

If the remainder is 0, divisibility holds. So the idea of remainder is already hiding inside the idea of divisibility.

A small example: divisors of 12

The positive divisors of 12 are

$$1, 2, 3, 4, 6, 12.$$

Each one divides 12 with remainder 0. This is a simple example, but it already points toward later questions:

• Which divisors do two numbers share?
• Which divisor is the greatest among them?
• How is divisibility tied to prime factorization?

Common mistakes

1. Introducing the notation too mechanically

Do not just memorize a | b. Read it as “a divides b.” Then connect it to the existence of an integer multiple.

2. Mixing divisibility with ordinary division

a | b is a relation. It does not mean the same thing as writing the fraction b/a.

3. Forgetting why remainder matters

The whole point is that divisibility means remainder 0. That is the bridge to the division algorithm and later to congruence.

Wrap-up

This post introduced the first language of number theory: divisibility, divisors, multiples, quotients, and remainders.

Next we make this more precise through the division algorithm, the standard form that organizes integer division.