The Rules of Divisibility
Number Properties is an important topic in math class and crucial for solving many problems. Number Properties are all about how numbers behave or interact with each other in different situations.
Learning and practicing number properties can be really helpful in your classes, exams, and even in everyday life. When you understand number properties, you can predict how numbers will behave and what kind of outcome to expect in a problem.
In this blog post, we will discuss the Rules of Divisibility. This is the third post in our Number Properties series, and if you haven't seen the first two, make sure to check them out so you can become a master in this foundational math topic.
The first two topics we covered were:
Stay tuned for our upcoming blogs, where we'll continue our discussion on number properties. We'll be exploring topics like:
Multiplication Math Facts
Knowing your multiplication math facts is extremely helpful in any math class. These facts serve as the foundation for building your math skills. Multiplication math facts come into play when dealing with factors, multiples, squares and square roots, and any other operation involving division or multiplication.
So, what exactly are multiplication math facts? They refer to the ability to quickly recall the multiplication table. For instance, can you answer the following questions without using a calculator?
4 x 3 = ?
3 x 4 = ?
12 ÷ 4 = ?
12 ÷ 3 = ?
How did you do? Were you able to answer all four questions easily? If not, don't worry! Keep practicing these multiplications and divisions because these are useful in all types of math.
There are various ways to practice your math facts. One method is by filling in a 12x12 multiplication table. Start by writing down as many multiplications as you know, and don't worry about the time it takes or the number of empty spaces left. Download our blank multiplication sheet to practice.
Once you're done, take a different colored pen or pencil and fill in the remaining products or correct any mistakes. Finally, use a third color to make a square around the products that are perfect squares. Repeat this activity until you can confidently complete the table.
Another way to study math facts is by creating flashcards on a computer or physically. It's helpful to have two sets of flashcards, one for multiplication and one for division. You can also use different colors to distinguish between the multiplication and division facts.
Understanding and recalling your multiplication math facts can simplify operations. However, as numbers get larger, you'll likely need other methods for division. Long division or the use of calculators can always be helpful. But, with these divisibility rules, we can start predicting whether a number will be evenly divisible by another.
Rules of Divisibility
So, what does evenly divisible mean? When a number is evenly divisible by another number, it means there is no remainder when you divide the first number (the divisor) by the second number (the dividend). In other words, the division can be done without any fractions or remainders, and an integer will be the result.
Now, let's dive into the divisibility rules. These rules can help us when solving division problems by telling us whether a number is evenly divisible by another number. Remember, these rules don't tell us the exact quotient; they only help us determine whether there will be a remainder in the division problem.
Below are the rules from numbers 1 to 10. At the end of this post, you can also download a Divisibility Reference Sheet!
Rules of Divisibility For Numbers 1 to 10
1 - Every number can be divided evenly by 1.
2 - All even numbers can be divided evenly by 2.
Is 854 evenly divisible by 2?
Since 854 ends in the digit 4, which is even, it is evenly divisible by 2.
(854 ÷ 2 = 427)
3 - If the sum of the digits in a number is divisible by 3, then the number itself is divisible by 3.
Is 432 evenly divisible by 3?
Add up the digits: 4 + 3 + 2 = 9, which is divisible by 3.
Therefore, 432 is evenly divisible by 3.
(432 ÷ 3 = 144)
Is 815 evenly divisible by 3?
Add up the digits: 8 + 1 + 5 = 14, and 14 is not divisible by 3, so 815 is not divisible by 3.
(815 ÷ 3 = 271.66)
4 - If a number can be divided by 2 twice, or if the last two digits of a number can be divided by 4, then the whole number can be divided by 4.
Is 84 evenly divisible by 4?
Dividing 84 by 2 gives us 42, which is even and can be divided by 2 again.
Therefore, 84 is evenly divisible by 4.
(84 ÷ 4 = 21)
Is 980 evenly divisible by 4?
We can look at the last two digits, 80.
Then number 80 can be divided by 4, so the whole number can be divided by 4
(980 ÷ 4 = 245)
5 - If a number ends in 0 or 5, it is divisible by 5.
Is 765 evenly divisible by 5?
Since the last digit is 5, 765 is evenly divisible by 5.
(765 ÷ 5 = 153)
6 - If a number is divisible by both 2 and 3, then it is evenly divisible by 6.
Is 432 evenly divisible by 6?
Since 432 is an even number and divisible by 2 (as we learned in rule 2),
and the sum of its digits (4 + 3 + 2 = 9) is divisible by 3 (as we learned in rule 3), therefore, it is evenly divisible by 6.
(432 ÷ 6 = 72)
7 - To determine if a number is evenly divisible by 7, double the last digit and subtract it from the number made by the other digits. If the result is divisible by 7, then the original number is also divisible by 7.
Is 672 evenly divisible by 7?
Doubling the last digit, 2, gives us 4.
Subtracting this from the number made by the other digits, 67, we get 63, which is evenly divisible by 7.
Therefore, 672 is evenly divisible by 7.
(672 ÷ 7 = 96)
8 - If a number can be divided by 2 three times, or if the last three digits of a number can be divided by 8, then the whole number can be divided by 8.
Is 256 evenly divisible by 8?
Dividing 256 by 2 gives us 128, which can be divided by 2 again to give us 64.
Since 64 is even and can be divided by 2 one more time, 256 is evenly divisible by 8.
(256 ÷ 8 = 32)
9 - If the sum of the digits in a number is divisible by 9, then the number itself is divisible by 9.
Is 8,154 divisible by 9?
Add up the digits: 8 + 1 + 5 + 4 = 18, which is divisible by 9.
Therefore, 8,154 is evenly divisible by 9.
(8,154 ÷ 9 = 906)
10 - If a number ends in 0, then it is divisible by 10.
Is 980 evenly divisible by 10?
Since the last digit is 0, 980 is evenly divisible by 10.
(980 ÷ 10 = 98)
Conclusion
By understanding and using these Rules of Divisibility, we can predict divisibility regardless of the number we're dealing with. Learning these rules, along with practicing our multiplication math facts, helps us understand how numbers interact with each other.
It's important to remember that these rules are based on prime factors, and understanding number properties builds a strong foundation for our math skills. Number properties continue to build upon each other and are essential for developing a solid math background.
Keep up the great work, and stay tuned for more blogs on number properties!
