Division, along with addition, subtraction, and multiplication, is one of the four basic operations of mathematics. Divisibility is a component of division that refers to the capacity of an integer to be divided.

Whether you love division or hate it, we all use division and divisibility in our everyday lives. Understanding how to equally divide units has been used as the basis for time structures, money, labour, construction, etc. since time immemorial.

So what exactly is divisibility? What is its significance in maths? What is its significance in the real world? And what tips and tricks are there to make the process of working out divisibility easier?

We are going to explore all this and more as we jump into answering the question, what is divisibility?

The term "divisibility" refers to the capacity of a number to be split into equal parts without a remainder. Some numbers are more divisible than others and if they are more divisible than any number that precedes it, they are called "highly composite numbers".

Divisibility has captured the intrigue of writers, philosophers, and mathematicians for millennia and some even saw it as key to the running of a healthy society!

### Divisibility terminology

As we discuss divisibility, it is important to understand some of the technical terminologies that are used by mathematicians.

• The dividend. The dividend is the number that is being divided. For example, in the question 16 ÷ 4, the dividend is 16.
• Divisor. The divisor is the number that we are dividing by. So in the same example of 16 ÷ 4, the divisor is 4.
• Quotient. The quotient is the solution to a division question. 16 ÷ 4 = 4. So 4 is the quotient.
• Remainder. The remainder is what is left over after performing a calculation to find an integer.
• Prime numbers. Prime numbers are whole numbers that can only be divided by 1 and themselves. Prime numbers include: 2, 5, 7, 11, 13, 17, 19...
• Composite numbers. Composite numbers are whole numbers that can be divided by more than just 1 and themselves.
• Integer. An integer is just another word for a whole number. 3 is an integer, 3.5 is not.
• Quotition. Quotition is a form of division that considers how many parts of a whole there are.
• Partition. Partition is a form of division that asks how big is each part that forms the whole.

Understanding division is, of course, the key to understanding divisibility.

Division is the calculation of how many times one number is contained within another. For example, there are 4 lots of 5 in 20. Therefore, 20 ÷ 5 = 4.

All integers are divisible by at least 1 and themselves. All even integers are divisible by at least 1, themselves, and 2. Numbers that can only be divided by 1 and themselves are known as prime numbers.

Prime numbers are an important part of mathematics and have been written about by mathematicians dating all the way back to the Ancient Egyptians and the Ancient Greeks.

There is no pattern or equation for finding new prime numbers, which makes their discovery highly extraordinary. Every new prime number discovered since the 1950s has been found by a computer. The discovery of new primes is also rewarded with prizes worth tens of thousands of dollars.

Prime numbers are considered to be so fundamental to the way that maths works that when the physicist and philosopher Carl Sagan wrote his novel Contact, he had the initial form of communication sent from the aliens as a string of prime numbers.

Highly composite numbers, also known as "anti-primes", are positive integers with more divisors than any of the smaller integers that precede them.

For example, 24 can be divided by 24, 12, 8, 6, 4, 2, 1. No other number up to 24 has that many divisors, therefore 24 is a highly composite number.

Highly composite numbers have fascinated mathematicians and philosophers for thousands of years.

In his dialogue Laws, Plato wrote that the highly composite number 5040 is a very useful number for dividing large things - such as citizens or land - into smaller parts.

He noted that 5040 can be divided by all the numbers from 1 to 12, excluding 11. Plato so was insistent on the importance of the number 5040, that some writers have since argued that he believed it to be a fundamental component to the success of a city.

Although Plato would not have called 5040 "highly composite", mathematicians have since noted his musings to be an early example of the concept.

Not only does divisibility play a key role in maths, but we also all use it in our day-to-day lives.

For example, there are 24 hours in a day and have been since Ancient Egyptian times. One of the many reasons that 24 was chosen as the base number for a day, was that it is a highly composite number.

As we saw earlier, 24 has a large number of divisors and can, therefore, be split into multiple different chunks. Day and night can roughly equate to 12 hours each. Morning and afternoon - half a day each - are therefore 6 hours. And early morning / late morning and early afternoon / late afternoon both equal 3 hours.

Of course, these are all approximations that depend on where you are in the world and what time of year it is, but the fact remains that the high divisibility of 24 makes it a very useful number for splitting time into chunks.

Divisibility is also used for money. For example, we know that there are 100 pennies in a pound because 100 ÷ 1 = 100.

Similarly, a pound can be split into two 50ps, five 20ps, or twenty 2ps. And £10 can be broken into two £5 notes, five £2 coins, ten £1 coins, twenty 50ps, etc. We know all of this because of our familiarity with divisibility.

UK money wasn't always so divisible. Before decimalisation in 1971, the British pound was made up of 20 shillings and a shilling was made up of 12 pence. So there was a total of 240 pence to the pound.

You can see why the system changed to numbers that allowed for far simpler divisibility!

### Divisibility rules

Divisibility rules are tips and tricks that can make working out whether one integer is divisible by another a lot simpler.

The first and simplest divisibility rule is one we have already covered: every positive integer is divisible by 1 and itself.

So, now we have covered the obvious one, let's find out some rules for other lower integers.

#### 2

This is another nice and easy one. The divisibility rule of 2 is simply that all numbers that end in an even digit - i.e. all even numbers - are divisible by 2.

So if you look at the final digit of a number and it ends in a 0, 2, 4, 6, or 8, then you automatically know it is divisible by 2.

#### 3

A number is always divisible by 3 if the sum of its digits totals a number divisible by 3. This is less complicated than it sounds.

For example, let's take the number 256. The sum of the digits in 256 is:

• 2 + 5 + 6 = 13

Is 13 divisible by 3?

No.

So 256 isn't either.

However, the sum of the digits of the number 255 is:

• 2 + 5 + 5 = 12

Is 12 divisible by 3?

Yes.

So 255 is also divisible by 3.

#### 4

The 4 divisibility rule is similar to the rule for 2. However, when it comes to 4, you need to check if the number formed by the last two digits of the integer is divisible by 4, rather than whether the last digit is divisible by 2.

For example, we know that the number 27,832 is divisible by 4 because the last two digits, 32, form a number that is divisible by 4 with no remainder.

32 ÷ 4 = 8

Therefore 27,832 is divisible by 4.

#### 5

The rule for 5 is simple. Any integer that ends in 0 or 5 is divisible by 5.

#### 6

Any number that is divisible by both 2 and 3 is also divisible by 6.

So first, you should check to see if the number is divisible by 2 - i.e. does it end with 0, 2, 4, 6, 8?

If it is, you should next check to see if it is divisible by 3 - i.e. do the digits add up to a number that is divisible by 3?

For example, is 6,102 divisible by 6?

• Does it end in an even number? Yes, so it is divisible by 2.
• Do the digits add up to a multiple of 3?
• 6 + 1 + 0 + 2 = 9
• 9 is divisible by 3
• Therefore, 6,102 is divisible by 6.

#### 7

The divisibility rule for 7 is a bit trickier, but it's simple enough to use with three or four-digit numbers.

First, you need to double the final digit of the integer. Then you subtract this number from the remaining digits in the rest of the number. If the number obtained is either 0 or an integer that is divisible by 7, then the original integer was also divisible by 7.

For example, is 217 divisible by 7?

• We begin by doubling the final digit, which is 7.
• 7 x 2 = 14.
• We then subtract that number from the remaining digits, which is 21.
• 21 - 14 = 7.
• 7 is divisible by 7, therefore 217 is also divisible by 7.

#### 8

A number is divisible by 8 if the last three digits are divisible by 8. This is a good rule for bigger numbers, though it does require having a solid grasp of your 8 times tables up to three-digit numbers.

For example, the number 469,832 is divisible by 8 because 832 is divisible by 8.

#### 9

The rule for 9 is similar to the rule for 3. A number is divisible by 9 if the sum of all the digits is divisible by 9. If we think of the 9 timetables:

• 9, 18, 27, 36...
• 3 + 6 = 9
• 2 + 7 = 9
• 1 + 8 = 9

Then, if we take a bigger number, the same rule applies.

• Is 27, 963 divisible by 9?
• 2 + 7 + 9 + 6 + 3 = 27
• You can then either observe that 27 is divisible by 9 or return to what we did earlier by doing...
• 2 + 7 = 9

#### 10

10 is the easiest of the lot. If a number ends with 0, then it is divisible by 10.