Divisors of a Number

Imagine you have two numbers, $ a $ and $ b $, where $ b \neq 0 $. We say that $ b $ is a divisor of $ a $ if dividing $ a $ by $ b $ results in no remainder. In other words, $ b $ divides $ a $ exactly, with nothing left over. $$ a = q \cdot b $$ where $ q $ is a natural number representing the quotient when $ a $ is divided by $ b $.

So, if there exists an integer $ q $ such that $ a = q \cdot b $, we can conclude that $ b $ is a divisor of $ a $.

It’s important to note that b cannot be zero $ b \neq 0 $.

In fact, division by zero is undefined: in mathematics, dividing by zero isn’t possible. However, any non-zero number $ n $ is a divisor of zero because $ 0 \div n $ results in both quotient and remainder being zero.

For example, $ 3 $ is a divisor of $ 12 $ because $ 12 = 3 \cdot 4 $, and the division $ 12 \div 3 $ leaves a remainder of zero.

example of division

There are several ways to express that $ b $ is a divisor of $ a $:

$ a $ is a multiple of $ b $
$ b $ divides $ a $
$ a $ is divisible by $ b $

These expressions are interchangeable and all describe the same divisibility relationship between $ a $ and $ b $.

There’s a strong link between divisors and multiples: if $ b $ is a divisor of $ a $, then $ a $ is a multiple of $ b $, and the reverse is also true.

For instance, knowing that $ 12 $ is a multiple of $ 3 $ instantly tells us that $ 3 $ is a divisor of $ 12 $, and vice versa.

example of the relationship between divisors and multiples

This relationship is essential in mathematics as it directly connects division and multiplication, giving us deeper insight into the structure and properties of numbers.

Remember, unlike multiples, the divisors of a number are finite.

To find them, you need to identify all the numbers that divide the given number exactly.

For instance, the divisors of $ 12 $ are:

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

These numbers divide $ 12 $ without leaving a remainder and represent all possible divisors of $ 12 $.

Why is this useful? Knowing the divisors of a number is essential for solving complex arithmetic problems and for tackling advanced concepts like prime factorization or finding the greatest common divisor (GCD).

Visualizing Divisors

To illustrate a number’s divisors, you can use a divisor lattice.

This is a graphical tool that helps you visualize a number’s divisors and their hierarchical relationships.

In a divisor lattice, larger numbers are connected to smaller ones based on divisibility.

For example, in the divisor lattice of $ 12 $, the number $ 12 $ is connected to all its divisors (1, 2, 3, 4, 6), illustrating that each one divides $ 12 $ exactly.

divisors of the number 12

Understanding the lattice. At the top of the lattice is the number 12, the main number, divided by all other divisors. The second level includes 4 and 6, which are direct divisors of 12. The third level shows 2 and 3, which also divide 12 and are divisors of 4 and 6 as well, placing them one level lower. Finally, the lowest level always has the number 1, which divides every number.

The divisor lattice helps you see not only a number’s divisors but also how they relate to each other.