Definition of Divisibility Tests

A divisibility test is an easy way of finding out whether a given integer is divisible by a fixed divisor without executing the division, usually by examining its digits.
We say that a number is divisible by another number if the result of the division is a whole number”.
\(18\) is divisible by \(6\), because \(18 \div 6 = 3\)
But, \(17\) is not divisible by \(6\) because the result is not a whole number.
Since every number is not completely divisible by every other number, such numbers leave a remainder other than zero. Nevertheless, certain rules help us determine the actual divisor of a number just by considering the digits of the number.
The divisibility tests from \(2\) to \(13\) are explained here.

Divisibility Tests and Methods

We can quickly remember divisibility tests for numbers such as \(2\), \(3\), \(4\), and \(5\). But the divisibility tests for \(7\), \(11\), and \(13\) are a bit difficult, and for this reason, there is a need to understand them in detail.

Divisibility Test for \(2\)

If the last digit of the number is \(2\), \(4\), \(6\), \(8\), or \(0\), then we can say that the number is divisible by \(2\).
Example: The last digit of \(108\) is \(8\), it is divisible by \(2\). So, \(\frac{ {108}}{2} = 54\)

Divisibility Test for \(3\)

The divisibility test for \(3\) says that a number is completely divisible by \(3\) if the sum of the digits of the number are divisible by \(3\) or is a multiple of \(3\).
For example, consider two numbers, \(406\) and \(201\):
To check if \(406\) is divisible by \(3\) or not, find the sum of the digits.
\(4 + 0 + 6 = 10.\) Since the sum is \(10\) which not divisible by \(3\), \(406\) is not divisible by \(3\).
Check \(201\) by summing its digits: \(2 + 0 + 1 = 3\), since \(3\) is a multiple of \(3\), then \(201\) is divisible by \(3\).

Divisibility Test for \(4\)

If the last two digits of a given number are divisible by \(4\), we can say that the given number is divisible by \(4\).
For example: Consider two numbers, \(2516\) and \(9506\).
The last two digits of the number \(2516\) are \(16\). Since \(16\) is divisible by \(4\), \(2516\) is also divisible by \(4\).
\(9506\) is not divisible by \(4\) because the last two digits, \(06\), are not divisible by \(4\).

Divisibility Test for \(5\)

Any numbers ending with the digit as \(0\) or \(5\) are divisible by \(5\).
For example, in \(200\) the last digit is \(0\). So, \(\frac{ {200}}{5} = 40\) and \(9005\) is divisible by \(5\) as the unit digit is \(5\).
\(\frac{ {9005}}{5} = 1801\)

Divisibility Test for \(6\)

If the last digit of the given number is an even number or zero and divisible by \(2\) and the sum of the digits is divisible by \(3\), then the given number is divisible by \(6\). In other words, any number which is divisible by both \(2\) and \(3\) are divisible by \(6\).
For example, \(960\) is divisible by \(2\) because the last digit is \(0\).
Also, the sum of the digits is \(9 + 6 + 0 = 15\), which is also divisible by \(3\).
Therefore, \(960\) is divisible by \(6\).

Divisibility Test for \(7\)

To check whether a number is divisible by \(7\), subtract twice the unit digit from the remaining and check whether it is divisible by \(7\). If the doubled number is larger than the number formed by the other digits, then subtract other number from the doubled number. Continue the above process till you get a simple number.
Consider a number \(1073\). To check whether the number is divisible by \(7\) or not, remove the last number \(3\) and multiply it by \(2\), which becomes \(6\). Then subtract \(6\) from the remaining number \(107\), therefore \(107 – 6 = 101\).
Repeat the process. We have \(1 \times 2 = 2\), and the remaining number is \(10 – 2 = 8\). Since \(8\) is not divisible by \(7\), therefore the number \(1073\) is also not divisible by \(7\).
Example: To check the divisibility of \(252\), we must remove the last \(2\) and multiply with \(2\).
\( \Rightarrow 2 \times 2 = 4\). Now we subtract \(4\) from remaining number \( \Rightarrow 25 – 4 = 21\).
Clearly, \(21\) is divisible by \(7\). So, \(252\) is divisible by \(7\).

Divisibility Test for \(8\)

The divisibility test for \(8\) states that if the number is divisible by \(8\), its last three digits must be divisible by \(8\).
Example: Consider \(25152\). The last three digits of \(25152\) is \(152\), which is divisible by \(8\). So the given number \(25152\) is divisible by \(8\).

Divisibility Test for \(9\)

The divisibility test for \(9\) is similar to the divisibility test for \(3\). Thus, if the sum of the digits of a given number is divisible by \(9\), then the number is also divisible by \(9\).
Example: Consider \(4320\). The sum of the digits of \(4320\) is \(4 + 3 + 2 + 0 = 9\), which is divisible by \(9\). So, \(4320\) is divisible by \(9\).

Divisibility Test for \(10\)

The divisibility test for \(10\) states that any number whose last digit is zero is divisible by \(10\).
For example, \(40\), \(60\), \(9000\), \(4033000\) are divisible by \(10\).

Divisibility Test for \(11\)

This rule states that in a given number, if the difference of the sum of alternative digits is \(0\) or divisible by \(11\), then the number is divisible by \(11\).
For example, to check whether number \(6142\) is divisible by \(11\) or not, the procedure is:
The sum of alternative digits of each group is: \(6 + 4 = 10\) and \(1 + 2 = 3\)
Therefore, \(10 – 3 = 7\), which is not divisible by \(11\). Therefore \(6142\) is not divisible by \(11\).
There are some other conditions to check the divisibility by \(11\).
Method 1: If the number of digits of a given number is even, add the first digit and subtract the last digit from the number.
Example: \(2794\)
Number of digits \(= 4\)
Now, \(79 + 2 – 4 = 77\) is a multiple of \(11\).
So, \(2794\) is divisible by \(11\).
Method 2: If the number of digits of the given number is odd, then subtract the first and the last digits from the remaining number.
Example: \(62801\)
Number of digits \(= 5\)
Now, \(280 – 6 – 1 = 273\) is not divisible \(11\)
So, \(62801\) is not divisible by \(11\).
Method 3: Form the groups of two digits from the right end digit to the left end of the given number and find the sum of the resultant groups. If the sum is divisible by \(11\), then the given number is divisible by \(11\).
Example: \(4852 \Rightarrow 48 + 52 = 100\), which is not divisible by \(11\).
So, \(4852\) is not divisible by \(11\).
Method 4: Subtract the last digit of the given number from the rest of the number. If the outcome is a multiple of \(11\), then the given number will be divisible by \(11\).
Example: \(286 \Rightarrow 28 – 6 = 22\), which is a multiple of \(11\). So, \(286\) is divisible by \(11\).

Divisibility Test for \(12\)

If the given number is divisible by both \(3\) and \(4\). Then the given number is divisible by \(12\).
Example: \(9280\)
Sum of the digits of \(9280\): \(9 + 2 + 8 + 0 = 19\) which is not divisible by \(3\)
So, \(9280\) is not divisible by \(12\).

Divisibility Test for \(13\)

To test whether a number is divisible by \(13\), the last digit is multiplied by \(4\) and added to the remaining number until we get a two-digit number. If the two-digit number is divisible by \(13\), then the given number is also divisible by \(13\).
Example: \(1365\)
Now, \(136 + \left({5 \times 4} \right) = 136 + 20 = 156\)
Also, \(15 + \left({6 \times 4} \right) = 15 + 24 = 39\), which is divisible by \(13\).
So, \(1365\) is divisible by \(13\).

Solved Examples – Divisibility Tests

Q.1. Check if \(109816\) is divisible by \(8\).
Ans: Given: \(109816\)
Last three digits of \(109816\) is \(816\). Since \(816 \div 8 = 102\). The given number \(109816\) is divisible by \(8\).

Q.2. Check if \(1312\) is divisible by \(4\).
Ans: Given: \(1312\)
Last two digits of \(1312\) is \(12\) which is divisible by \(4\). So \(1312\) is divisible by \(4\).

Q.3. Check if \(672\) is divisible by \(7\).
Ans: Given: \(672\)
Multiply the last digit of the given number by \(2 \Rightarrow 2 \times 2 = 4\)
Subtract it from the remaining number \( \Rightarrow 67 – 4 = 63\)
Now, \(63\) is divisible by \(7\). So, the given number \(672\) is divisible by \(7\)

Q.4. Check if \(3729\) is divisible by \(11\).
Ans: Given: \(3729\)
If the number of digits of a given number is even, add the first digit and subtract the last digit from the number.
\(3729 \Rightarrow 72 + 3 – 9 = 66\), which is divisible by \(11\). So, \(3729\) divisible by \(11\).

Q.5. Check if \(2795\) is divisible by \(13\).
Ans: Given: \(2795\)
\(2795 \to 279 + \left({5 \times 4} \right) = 299\)
Now, \(299 \to 29 + \left({9 \times 4} \right) = 29 + 36 = 65\), which is divisible by \(13\).
So, \(2795\) is divisible by \(13\).

Summary

In this article, we have learned about divisibility tests’ meaning and learnt the different methods to test the divisibility of the given numbers by \(1\) to \(13\) and solved some examples.

Frequently Asked Questions (FAQs) – Divisibility Tests

Q.1. Write down the divisibility rule for \(9\).
Ans: The divisibility test for \(9\) is similar to the divisibility test for \(3\). If the sum of the digits of a given number is divisible by \(9\), then the number is also divisible by \(9\).

Q.2. What is the divisibility rule for \(2\) and \(5\)?
Ans: If the last digit of the number is \(2\), \(4\), \(6\), \(8\), or \(0\). Then we can say that the number is divisible by \(2\).
Any numbers ending with the digit as \(0\) or \(5\) are divisible by \(5\).

Q.3. What is the divisibility rule for \(7\)? Give an example.
Ans: The divisibility rule for \(7\) is mentioned below:
Remove the last digit from the given number and multiply it by \(2\).
Subtract the resultant from the remaining number.
If the outcome of the above step is a multiple of \(7\). Then the given number is divisible by \(7\).
Example: To check the divisibility of \(133\), we must remove the last \(3\) and multiply with \(2\)
\( \Rightarrow 3 \times 2 = 6\). Now we subtract \(6\) from the remaining number \( \Rightarrow 13 – 6 = 7\).
Clearly, \(7\) is divisible by \(7\). So, \(133\) is divisible by \(7\).

Q.4. Write the divisibility rule for \(6\)?
Ans: Any number which is divisible by both \(2\) and \(3\) are divisible by \(6\).

Q.5. What is the divisibility rule for \(13\)?
Ans: To test whether a number is divisible by \(13\), the last digit is multiplied by \(4\) and added to the remaining number until we get a two-digit number. If the two-digit number is divisible by \(13\), then the given number is also divisible by \(13\).

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