# Category: 因数と倍数

## Divisibility of Integers

When we talked about division of integers, we assumed that result of the division is an integer number.

From another side, when we talked about division with remainder, we discovered that result of the division is not always an integer number.

## Even Numbers (Integers)

A number (integer) is even if it is divisible by 2.

Example 1. Determine whether 10 is an even number.

10 is even because $$$\frac{{10}}{{2}}=5$$$ and 5 is an integer number.

Another example.

Example 2. Determine whether 15 is even.

## Odd Numbers (Integers)

Number (integer) is odd if it is NOT divisible by 2.

Alternatively, number is odd if it is not even.

Example 1. Determine whether 24 is odd number.

24 is even because $$$\frac{{24}}{{2}}={12}$$$ and 12 is integer number. Thus, 24 is not odd number.

## Divisibility Rules

In general it is hard to determine from first glance whether one number is divisible by another. We need to perform division.

However, in some cases, we can determine whether one number divides another without dividing them by applying divisibility rules.

## What are Factors and Multiples

Let's see what are factors and multiples.

When we talked about divisibility of integers, we said that number $$${a}$$$ is divisible $$${b}$$$ if $$${c}=\frac{{a}}{{b}}$$$ is integer number.

If $$${a}$$$ is divisible by $$${b}$$$, then $$${b}$$$ is a factor of $$${a}$$$ and $$${a}$$$ is multiple of $$${b}$$$.

## Integer Factorization

When we talked about factors and multiples we learned how to find factors of number.

This allows to write number as product of its factors.

Recall that factors of 12 are 1,2,3,4,6,12.

So, we can write that $$${12}={1}\cdot{12}$$$, $$${12}={2}\cdot{6}$$$ and $$${12}={3}\cdot{4}$$$.

## What is a Prime Number

Prime number is an integer number greater than 1 that has only two factors: 1 and itself.

For example, 5 is prime number because it has no other factors except 1 and 5.

By convention 1 is not prime, so prime numbers start from 2.

## Composite Numbers

Composite number is a number, that is not prime.

For example, 6 is composite number because its has other factors except 1 and 6, namely, 2 and 3.

By convention 1 is not composite.

This means that 1 is neither prime, nor composite.

## How do you do Prime Factorization

When we talked about integer factorization we noticed that integer number can have more than one factorization.

For example, $$${18}={1}\cdot{18}={2}\cdot{9}={3}\cdot{6}={2}\cdot{{3}}^{{2}}$$$.

However, if we use only prime numbers as factors than such factorization is unique (ignoring order of factors).

## Greatest Common Divisor (GCD)

Suppose we are given two numbers 18 and 24.

Let's find their factors.

18: 1,2,3,6,9,18.

24: 1,2,3,4,6,8,12,24.

As can be seen some factors are same for both numbers (they are in bold). These numbers are called common factors (divisors) of 18 and 24.

## Least Common Multiple (LCM)

Suppose we are given two numbers 18 and 24.

Let's find some of their multiples.

18: 18,36,54,72,90,108,126,144,...

24: 24,48,72,96,120,144,168,...

As can be seen some factors are same for both numbers (they are in bold: 72 and 144). These numbers are called common multiples of 18 and 24.