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when is number divisible by 8 ?

# When is a Number Divisible by 8?

In the fascinating world of mathematics, the divisibility rules serve as a quick and efficient way to determine whether a number can be divided evenly by another without performing the actual division. Among these rules, the divisibility rule for 8 is especially useful and straightforward for larger numbers. In this article, we will delve into the divisibility rule for 8, explore examples, and illustrate how these rules can simplify your mathematical calculations.

## Understanding the Divisibility Rule for 8

The divisibility rule for 8 states that a number is divisible by 8 if:

– The last three digits of the number are zeros, or
– The number formed by the last three digits of the number is divisible by 8.

This rule is based on the fact that any number can be expressed in the form of (1000a + b), where (a) represents the number formed by the thousands, millions, etc., part of the number, and (b) represents the last three digits. Since 1000 is already divisible by 8, the divisibility of the entire number by 8 is entirely dependent on the last three digits, (b).

## Practical Examples

Let’s explore some practical examples to illustrate this rule:
– **Example 1:**
– **Number:** 1,848
– **Last Three Digits:** 848
– **Is 848 Divisible by 8?** Yes, because (848 div 8 = 106).
– **Conclusion:** Since the last three digits are divisible by 8, 1,848 is also divisible by 8.

– **Example 2:**
– **Number:** 5,600
– **Last Three Digits:** 600
– **Is 600 Divisible by 8?** Yes, because (600 div 8 = 75).
– **Conclusion:** Since the last three digits are divisible by 8, 5,600 is also divisible by 8.

– **Example 3:**
– **Number:** 123,456
– **Last Three Digits:** 456
– **Is 456 Divisible by 8?** Yes, because (456 div 8 = 57).
– **Conclusion:** Since the last three digits are divisible by 8, 123,456 is also divisible by 8.

## Alternative Method: Using the Last Two Digits

Another method for checking the divisibility by 8 for numbers with less than three digits or for a quick mental check of your results, involves evaluating the hundreds digit and the last two digits of the number:

– If the hundreds digit is even and the last two digits form a number that is divisible by 8, then the number is divisible by 8.
– If the hundreds digit is odd, add 4 to the number formed by the last two digits, and if the new number is divisible by 8, the original number is divisible by 8.

Let’s apply this method to a few examples:
– **Example:**
– **Number:** 1,236
– **Hundreds Digit:** 3 (odd)
– **Last Two Digits:** 36 + 4 = 40
– **Is 40 Divisible by 8?** Yes, because (40 div 8 = 5).
– **Conclusion:** 1,236 is divisible by 8.

– **Example:**
– **Number:** 5,624
– **Hundreds Digit:** 6 (even)
– **Last Two Digits:** 24
– **Is 24 Divisible by 8?** Yes, because (24 div 8 = 3).
– **Conclusion:** 5,624 is divisible by 8.

## A Few More Examples:

– **Number:** 8,640
– **Last Three Digits:** 640 (divisible by 8)
– **Conclusion:** Hence, 8,640 is divisible by 8.

– **Number:** -77,184
– **Last Three Digits:** 184 (divisible by 8)
– **Conclusion:** Hence, -77,184 is divisible by 8.

– **Number:** 20,233,322,496
– **Last Three Digits:** 496 (divisible by 8)
– **Conclusion:** Hence, 20,233,322,496 is divisible by 8.

## Additional Insights

### Link to Other Rules: Factors of 8

When you check whether a number is divisible by 8, remember that 8 is a factor of 16, 24, 32, and so on. Additionally, 8 is a multiple of 4, meaning that any number divisible by 8 is also divisible by 4. Conversely, a number divisible by 8 is not necessarily divisible by 16 or higher multiples of 8, unless you verify the divisibility by these numbers separately.

### Practicing the Rule

Here are a few numbers and their divisible status according to the rule:
– **1,248:** (48 div 8 = 6) -> Divisible by 8.
– **3,456:** (456 div 8 = 57) -> Divisible by 8.
– **5,672:** (672 div 8 = 84) -> Divisible by 8.
– **7,890:** (890 div 8 = 111.25) -> Not Divisible by 8.

## Conclusion

Mastering the divisibility rule for 8, along with other divisibility tests for different numbers, enhances your mental arithmetic skills and can solve problems more quickly in mathematics and real-world applications. Remember, practice makes perfect, so keep experimenting with different numbers to check their divisibility by 8. Whether you’re dealing with straightforward calculations or tackling complex problems, these rules are your go-to tools to simplify the process.

Happy calculating!

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Feel free to reach out with any questions or if you need more examples!