Introduction of Divisibility Test
In the world of math, the Divisibility Test is like a super handy tool. It helps us quickly figure out if one number can be divided by another without leaving any remainders. In this detailed post, we’ll take a close look at how the Divisibility Test works. We’ll break down its rules and give you lots of tips to help you become at home in math.
Understanding the Divisibility Test
The Divisibility Test is basically a cool trick that helps us figure out if one number can be divided evenly by another, without actually doing the division. With some easy rules and tricks, we can quickly tell if numbers can be divided or not, which saves us a bunch of time and hassle when doing math.
The Fundamental Rules
Divisibility Test by 1
Every number is divisible by 1.
Divisibility Test by 2
A number is divisible by 2 if its last digit is even. For example, consider the number 468. Since the last digit is 8, which is even, 468 is divisible by 2.
Divisibility Test by 3
To determine if a number is divisible by 3, we sum its digits. If the resulting sum is divisible by 3, then the original number is also divisible by 3. For instance, let’s take the number 639. The sum of its digits is 6 + 3 + 9 = 18. Since 18 is divisible by 3, we conclude that 639 is divisible by 3.
Divisibility Test by 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, let’s consider the number 248. The number formed by its last two digits is 48, which is divisible by 4. Therefore, 248 is divisible by 4.
Divisibility Test by 5
A number is divisible by 5 if its last digit is either 0 or 5. Let’s examine the number 375. Since the last digit is 5, 375 is divisible by 5
Divisibility Test by 6
A number is divisible by 6 if it is divisible by both 2 and 3. For example, consider the number 462. Since 462 is divisible by 2 (last digit is even) and 3 (sum of digits is 4 + 6 + 2 = 12, which is divisible by 3), it is divisible by 6.
Divisibility Test by 7
Divisibility tests might sound tricky, but don’t worry. We’ve got an easy way to figure out if a number can be divided by 7. Just follow these simple steps:
Step 1: Start by taking the last digit of your number and multiply it by 2 .
Step 2: Subtract this doubled digit from the number made up of all the other digits.
Step 3: Now, here’s where the magic happens: Check if the difference you found in Step 2 is divisible by 7.
If it is, Your entire number is divisible by 7.
Let’s put it into practice with an example:
Take the number 294.
Step 1: Double the last digit (4 x 2 = 8).
Step 2: Subtract the result (8) from the number formed by the other digits (29 – 8 = 21).
Step 3: Check if 21 is divisible by 7. yes, it is.
So, we conclude that the given number, 294, is divisible by 7.
Note:For smaller numbers upto 4 digit we follow above test rule. But for larger numbers we have to some more calculations. For these calculation click here.
Divisibility Test by 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
For instance, let’s take the number 1,248. The number formed by its last three digits is 248, which is divisible by 8. Hence, 1,248 is divisible by 8.
Divisibility Test by 9
The rule for divisibility by 9 is similar to the divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.
Example: Consider 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9
Example: Consider 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9
Divisibility Test by 10
Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.
Example: 10, 20, 30, 1000, 5000, 60000, etc.
Divisibility Test by 11
If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.
i.e., Sum of digits in odd places – Sum of digits in even places = 0 or a multiple of 11
To check whether a number like 2143 is divisible by 11, below is the following procedure.
Group the alternative digits i.e. digits that are in odd places together and digits in even places together. Here 24 and 13 are two groups.
Take the sum of the digits of each group i.e. 2+4=6 and 1+3= 4
Now find the difference of the sums; 6-4=2
If the difference is divisible by 11, then the original number is also divisible by 11. Here 2 is the difference which is not divisible by 11.
Therefore, 2143 is not divisible by 11.
FAQ
Divisibility is the property of one number being divisible by another without leaving a remainder. In simpler terms, it means whether one number can be divided by another number evenly.
A divisor is a number that divides another number without leaving a remainder. For example, in the division 10 ÷ 2 = 5, 2 is the divisor.
A multiple is the result of multiplying a number by an integer (a whole number). For example, the multiples of 3 are 3, 6, 9, 12, etc.
There are specific rules and tests for divisibility by certain numbers. For example, a number is divisible by 2 if its last digit is even, and a number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 2 if its last digit is even.
A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
A number is divisible by 5 if its last digit is 0 or 5.
A number is divisible by 6 if it is divisible by both 2 and 3.
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
A number is divisible by 9 if the sum of its digits is divisible by 9.
A number is divisible by 10 if its last digit is 0.
- While there are specific rules for some numbers, there isn’t a single rule that applies to all numbers. However, understanding prime factorization can help determine divisibility for a wider range of numbers.
Divisibility rules are useful in various situations, such as simplifying fractions, determining if a number is prime, or checking if quantities can be divided evenly.
Yes, there are various shortcuts and tricks, such as quickly checking divisibility by 2, 5, or 10 based on the last digit, or using divisibility rules for smaller prime numbers to quickly determine divisibility for larger numbers.
Common mistakes include forgetting to apply the rules correctly, not considering the entire number, or misinterpreting the results of the divisibility tests.
Yes, while some rules are specifically for smaller numbers, understanding the principles behind divisibility can help in devising rules for larger numbers or using prime factorization to determine divisibilit
