Divisibility Rule of 5

The divisibility rule is a simple rule to determine whether a number can be divided by a certain number without leaving a remainder. The Divisibility Rule of 5 simplifies the process of determining whether a number can be divided by 5 without leaving a remainder.

This rule states that if the last digit of the number is either 0 or 5, then the number is divisible by 5.

For Example: Check if 145 is divisible by 5.
The last digit is 5.
Since the last digit is 5, 145 is divisible by 5.

By using this rule, we can easily check the divisibility of any number by 5. It helps to determine the divisibility by 5 without performing long division.

More Examples of Divisibility Rule of 5

We can easily determine whether a number is divisible by 5 by simply examining its last digit. A few Examples for the same are:

• 35: The last digit is 5, so 35 is divisible by 5.
• 80: The last digit is 0, so 80 is divisible by 5.
• 123: The last digit is 3, so 123 is not divisible by 5.
• 450: The last digit is 0, so 450 is divisible by 5.

Proof of Divisibility Rule of 5

Any number N can be expressed in the form of a sum of powers of 10. For example, the number 1234 can be written as:

N = 1⋅10³ + 2⋅10² + 3⋅10¹ + 4⋅10⁰

In general, for a number with digits d_n​, d_n-1,......d₁, d_0, the number can be written as:

N = d_n \cdot 10^n + d_{n-1} \cdot 10^{n-1} + \dots + d_1 \cdot 10^1 + d_0 \cdot 10^0

Notice that powers of 10 are always divisible by 5. Let's look at the first few powers of 10:

• 10⁰ = 1, not divisible by 5.
• 10¹ = 10, divisible by 5.
• 10² = 100, divisible by 5.
• 10³ = 1000, divisible by 5.

In fact, for all n ≥ 1, 10ⁿ is divisible by 5

Since all powers of 10 (except 10⁰) are divisible by 5, the only part of N that determines its divisibility by 5 is the last term, which is d₀. 10⁰ = d₀.

Thus, for the whole number N to be divisible by 5, the last digit d₀​ must be divisible by 5.

So the last digit should either be zero or 5.

Verification with Table of 5

The following are the numbers in table of 5 and their last digit.

5 : 5
10 : 0
15 : 5
20 : 0
25 : 5
30 : 0
35 : 5
40 : 0
45 : 5
50 : 0

Divisibility Rule of 5 for Large Numbers

The divisibility rule of 5 is particularly straightforward and remains effective regardless of the size of the number. The rule states that a number is divisible by 5 if its last digit is either 0 or 5. This test applies to both small and large numbers. We have to follow these simple steps:

• Identify the Last Digit: Look at the last digit of the number.
• Apply the Rule:
  • If the last digit is 0 or 5, the number is divisible by 5.
  • If the last digit is any other digit (1, 2, 3, 4, 6, 7, 8, 9), the number is not divisible by 5.

Let us consider an example to check the divisibility by 5 for larger numbers:

Example: Check the divisibility by 5 for the number 123,456,780.

Solution:

Given number: 123,456,780

Last digit of the number is 0

Hence 123,456,780 is divisible by 5.

Divisibility Rule of 5 and 10

As we have already studied about divisibility rule of 5 which states that a number is divisible by 5 if its last digit is either 0 or 5.

Similarly, Divisibility rule of 10 states that a number is divisible by 10 if its last digit is 0. This rule is also straightforward and easy to apply.

Let us take a few examples for the same:

Example 1: Check if the number 225 divisible by 5 and 10?

Solution:

Given Number: 225

• Last digit: 5
• Divisibility by 5: Yes, 25 is divisible by 5.
• Divisibility by 10: No, 25 is not divisible by 10.

Example 2: Check if the number 4730 divisible by 5 and 10?

Solution:

Given Number: 4730

• Last digit: 0
• Divisibility by 5: Yes, 4730 is divisible by 5.
• Divisibility by 10: Yes, 4730 is divisible by 10 also.

Hence, we can say that if the last digit of a number is zero(0), then it is divisible by both 5 and 10.

How to Find Numbers Divisible By 5?

Numbers divisible by 5 are those that end in either 0 or 5. This applies to numbers of any length and simplifies the process of checking divisibility. There are infinite numbers which are divisible by 5, however we can determine how many numbers are divisible by 5 between 1 and a given number 'n' by the formula:

Count = n/5

For Example: To calculate number between 1 and 100 we will use the above formula:

Count = n/5

Here n = 100

Count = 100/5 = 20

Hence there are 20 numbers between 1 and 100 that are divisible by 5.

Other Divisibility Rules:

Divisibility Rule of 5 - Solved Questions

Example 1: Is the number 2460 divisible by 5?

Solution:

Given number is 2460

• Identify the last digit of the number.
• The last digit of 2460 is 0.
• Since the last digit is 0, 2460 is divisible by 5.

Example 2: Use the divisibility rule of 5 to check the divisibility of 789 by 5?

Solution:

Given number: 789

• Identify the last digit of the number.
• The last digit of 789 is 9.
• Since the last digit is not 0 or 5, 789 is not divisible by 5.

Example 3: Is the number 5805 divisible by 5?

Solution:

Given number is 5805

• Identify the last digit of the number.
• The last digit of 5805 is 5.
• Since the last digit is 5, 5805 is divisible by 5.

Example 4: Find the largest 4-digit number divisible by 5.

Solution:

To find the largest 4-digit number divisible by 5, we need to consider the largest possible 4-digit number and then find the largest one that is divisible by 5.

The largest 4-digit number is 9999.

To find the largest 4-digit number divisible by 5, we need to find the largest multiple of 5 that is less than or equal to 9999.

The largest multiple of 5 less than or equal to 9999 is 9995.

So, the largest 4-digit number divisible by 5 is 9995.

Practice Questions on Divisibility Rule of 5

Here are some practice questions on the divisibility rule of 5 for you to solve:

Q1: Determine whether the following numbers are divisible by 5:

a) 420

b) 287

c) 875

d) 900

e) 333

Q2: Find the smallest 3-digit number divisible by 5.

Q3: How many numbers between 100 and 200 are divisible by 5?

Also Read: Practice Questions on Divisibility Rules

Conclusion

The rule for divisibility by 5 offers a straightforward method for quickly determining if a number can be divided by 5 without performing the actual division. By simply checking if a number ends in either 0 or 5, we can efficiently identify its divisibility. This simple rule not only simplifies calculations but also improves our understanding of number patterns and arithmetic properties.