# Divisibility rule of 5

Created by: Team Maths - Examples.com, Last Updated: June 10, 2024

## Divisibility rule of 5

The divisibility rule for 5 is a fundamental principle in mathematics that states a number is divisible by 5 if its last digit is either 0 or 5. This rule simplifies the process of determining divisibility without needing complex division, serving as a cornerstone in the study of integers and rational numbers. In algebra, understanding this rule aids in manipulating expressions and solving equations that involve multiplication and division. It applies exclusively to integers, as irrational numbers do not conform to divisibility rules. Mastery of this rule enhances efficiency in mathematical operations such as addition, subtraction, multiplication, and division, facilitating quicker mental calculations and problem-solving strategies.

## What is the Divisibility Rule of 5?

The divisibility rule for 5 states that a number is divisible by 5 if the last digit of the number is either 0 or 5. This simple criterion allows for quick assessments of divisibility, making it a valuable shortcut in basic arithmetic and number theory. It is applicable to any integer, facilitating easier calculations in tasks involving division and factorization.

## Proof of Divisibility Rule of 5

### Step 1: Consider the number 4370.

We will examine this specific number to determine if it is divisible by 5.

### Step 2: Observe the last digit of the number.

In the case of 4370, the last digit is ‘0’.

### Step 3: Apply the divisibility rule for 5.

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

### Step 4: Check the last digit of 4370.

The last digit, as noted, is 0.

### Step 5: Verify the rule through simple reasoning.

Since 0 is one of the digits that satisfies the condition of the divisibility rule for 5, we proceed to conclude the divisibility.

### Step 6: Conclude based on the rule.

Given that the last digit of 4370 is 0, by the rule of divisibility for 5, 4370 must be divisible by 5.

### Step 7: Perform a division check (optional for verification).

Dividing 4370 by 5 gives a quotient of 874, which is an integer with no remainder, confirming divisibility.

### Step 8: Summary and conclusion.

From this examination, we affirm that the number 4370 adheres to the divisibility rule for 5 due to its last digit being 0. Thus, 4370 is indeed divisible by 5.

## Divisibility Rule of 5 and 10

### Divisibility Rule for 5

A number is divisible by 5 if its last digit is either 0 or 5. This rule makes it easy to scan numbers and identify those divisible by 5. For example:

• Number 235: The last digit is 5, so it is divisible by 5.
• Number 420: The last digit is 0, so it is divisible by 5.

### Divisibility Rule for 10

A number is divisible by 10 if its last digit is 0. This rule is even simpler than the one for 5 because only numbers ending in 0 can be evenly divided by 10 without leaving a remainder. For example:

• Number 450: The last (rightmost) digit is 0, so it is divisible by 10.
• Number 567: The last digit is 7, so it is not divisible by 10.

## Divisibility Test of 5 and 6

A number is divisible by 5 if its last digit is either 0 or 5. This test is straightforward and easily observable by examining the final digit of the number. For example:

• Number 230: The last digit is 0, hence it is divisible by 5.
• Number 1475: The last digit is 5, hence it is divisible by 5.

## Divisibility Rule of 5 Examples

### Number 310

• Last Digit: 0
• Conclusion: Since the last digit is 0, the number 310 is divisible by 5.

### Number 647

• Last Digit: 7
• Conclusion: Since the last digit is not 0 or 5, the number 647 is not divisible by 5.

### Number 1825

• Last Digit: 5
• Conclusion: Since the last digit is 5, the number 1825 is divisible by 5.

### Number 2340

• Last Digit: 0
• Conclusion: Since the last digit is 0, the number 2340 is divisible by 5.

### Number 924

• Last Digit: 4
• Conclusion: Since the last digit is not 0 or 5, the number 924 is not divisible by 5.

## How can you tell if a large number is divisible by 5?

Regardless of the size of the number, check the last digit. If it is 0 or 5, then the number is divisible by 5.

## Does the divisibility rule for 5 apply to negative numbers?

Yes, the rule applies to all integers, positive or negative. The last digit of the number determines divisibility, not the sign.

## Can this rule be used to simplify fractions?

Yes, if both the numerator and denominator end in 0 or 5, you can divide both by 5 to simplify the fraction.

## Why does the divisibility rule for 5 work?

In our base-10 number system, any number ending in 0 or 5 is a multiple of 5, hence the rule holds.

## Is there a similar rule for other digits like 0 or 5?

Yes, for divisibility by 10, the number must end in 0. For divisibility by 2, the number must end in an even digit (0, 2, 4, 6, or 8).

## Are there any exceptions to the divisibility rule for 5?

No, there are no exceptions. If the last digit is 0 or 5, the number is divisible by 5 without exception.

## How does the divisibility rule for 5 help in mathematical calculations?

It speeds up processes such as factorization and simplification, especially in algebra and arithmetic operations.

## What happens if a number is mistakenly identified as divisible by 5?

Errors in identifying divisibility may lead to incorrect simplifications or solutions, especially in calculations involving multiple steps or equations.

## Can you give an example of a real-world application of the divisibility rule for 5?

In financial contexts, quick divisibility checks can help when calculating discounts, sales prices, or when tallying amounts that involve multiples of 5 dollars, simplifying mental math and reducing errors.

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