Dividing, multiplying, and subtracting are fundamental operations in maths. People use the process in everyday life. Students and academics use division in complex equations at the highest academic levels.
In this article, we'll look at the terminology associated with division and find out what the term 'divisible' means. Continue reading to learn about divisibility rules and how you can tell whether a number is divisible or not.
Divisible means that one number can be divided by another, and the result is a whole number (without any remainders). If Integer A can be divided by Integer B and result in another full number (Integer C), Integer A is divisible by Interger B. However, if Integer C has remainders, then Integer A is not divisible by Integer B.
For example, when 10 is divided by 5, it results in 2, which is a full number. This means that 10 is divisible by 2. However, when you divide 10 by 3, the result is 3 remainder 1. As this result has a remainder, 10 isn't divisible by 3.
You will know if a number is divisible by another if it results in a full integer when you divide the first number by the second. For example, when dividing 10 by 2, you know it is divisible because the end result is 5, which is a full number.
It's helpful to know your multiplication tables to work out whether one number is divisible by another. For example, you should know that 25 is divisible by 5 because 5 x 5 = 25. However, 26 isn't divisible by 5 because 26 doesn't feature in the 5 times tables.
You can also use factors to work out if one number is divisible by another. When one number is divisible by another, then its factors are also divisible by that number. For example, if a number is divisible by 12, the number is also divisible by the factors of 12 (2,3,4 and 6).
As detailed below, there are several rules to tell if one number is divisible by another. These techniques can help you identify whether numbers are divisible by a specific number (such as whether a number is divisible by 2 or 3). These techniques are especially helpful when dealing with larger numbers.
All numbers are divisible by 1 and 0.
If the number is even (ending in 0,2,4,6 or 8), it will be divisible by 2.
If, when added together, the digits in the number are divisible by 3, the whole number is divisible by 3. For example, 456 is divisible by 3 because 4 + 5 +6 = 15, which is divisible by 3. However, 284 isn't divisible because 2 + 8 + 4 = 14 isn't divisible by 3.
This also works for finding out if a number is divisible by 9, although the digits should result in a number divisible by 9 when added together. For example, 486 is divisible by 9 because 4 + 8 + 6 = 18, which is divisible by 9.
If a number is divisible by 2 and 3, it will also be divisible by 6.
A number ending in 0 or 5 is divisible by 5.
If the number's last two digits are divisible by 4, the whole number is divisible by 4. For example, 2212 ends in 12, which is divisible by 4. However, 2213 isn't divisible by 4 because 13 (the last two digits) isn't.
Below are examples of numbers that are divisible by other numbers and how they were calculated.
Numbers that are divisible by 3
There are 30 two-digit numbers that are divisible by 3. The largest of these is 99, and the smallest is 6 (aside from 3 itself). No prime numbers are divisible by 3 as they are only divisible by 1 and themselves. However, both odd and even numbers can be divisible by 3. For example, 6, 12 and 18 are even and divisible by 3, as are the odd numbers 15, 21 and 27.
All numbers divisible by 9 are also divisible by 3, as 3 is a factor of 9.
Numbers that are divisible by 4
22 two-digit numbers are divisible by 4. The largest of which is 96, which is 24 x 4. 100 is also divisible by 4, meaning you only have to look at the last two digits of any number to see if it is divisible by 4. The same 25 two-digit numbers appear at the end of every multiple of 4. For example, 32 is divisible by 4, as is 132, 232, 332 and so on. 68 is also divisible by 4, as are 668, 1,468 and 20,568.
Numbers that are divisible by 7
There are 13 two-digit numbers that are divisible by 7, the largest of which is 98 (14 x 7). To work out if a number is divisible by 7, you need to double the units digit. In the example of 105, this would be 5, so 5 x 2 = 10. Next, you need to see if the difference between this answer and the remaining digits of the original number are equal to 0 or multiples of 7. In the given example, the remaining digits are 10, so 10 - 10 = 0. This means that 105 is divisible by 7.
However, 905 isn't divisible by 7. This is because the difference between the doubled unit (5 x 2 = 10) and the remaining digits (90) is more than 0 and not a multiple of 7 (90 - 10 = 80).
An easy way to check what numbers are divisible by others is to pretend you are sharing apples amongst friends. For example, if you have 15 apples, you could give five friends three apples each. However, if you had a sixth friend and took turns giving each of them an apple, three of your friends would have three apples, and the remaining three friends would only have two apples each at the end.
You don't necessarily have to know how the apples will be divided straight away. Instead, you can hand out one apple to each of your friends until all 15 apples have been shared out. If you were trying to replicate this on a piece of paper, you could draw five boxes and put a circle in each box until you have shared the 15 dots equally. You can then count how many dots are in each box.
Another way to think about divisibility is through slices of cake. If you had four people, you could slice the cakes into quarters, eighths or twelfths so that everyone had one, two or three slices each. You wouldn't slice the cake into ten pieces, though, because all eight people would have one slice, and there would be two remaining slices that can't be evenly shared.
Divisible means that one number results in a full number (with no remainders) when it is divided by another number. All numbers are divisible by 0 and 1. Various other rules can tell you if a number is divisible by specific numbers. For example, a number ending in 0 or 5 will always be divisible by 5. In addition, an even number will always be divisible by 2.
If Integer A is divisible by Integer B, Integer A will also be divisible by the factors of Integer B. For example, a number that is divisible by 10 will automatically be divisible by 2 and 5 as well, as they are the factors of 10.