how to check if a number is divisible by 24

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21-01-2025

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Divisibility rules are handy shortcuts for determining if a number is divisible by another number without performing the actual division. Knowing these rules can save you time and effort in various mathematical contexts. This article will explore how to efficiently determine if a number is divisible by 24.

Understanding the Divisibility Rule for 24

There isn't a single, standalone divisibility rule for 24 like there is for some smaller numbers (e.g., divisibility by 2 means the last digit is even). Instead, we leverage the prime factorization of 24 to create a simple test.

The prime factorization of 24 is 2³ x 3. This means that a number is divisible by 24 if it's divisible by both 8 (2³) and 3. Therefore, to check if a number is divisible by 24, we need to apply the divisibility rules for both 8 and 3.

Step-by-Step Guide: Checking Divisibility by 24

Here's a clear, step-by-step process:

Step 1: Check for Divisibility by 3

• The Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

• Example: Let's check if 72 is divisible by 3. 7 + 2 = 9, and 9 is divisible by 3. Therefore, 72 is divisible by 3.

Step 2: Check for Divisibility by 8

• The Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8. For numbers with fewer than three digits, check if the entire number is divisible by 8.

• Example: Let's check if 72 is divisible by 8. Since 72 has only two digits, we check if 72 itself is divisible by 8. It's not (72 / 8 = 9).

Let's try another number: 1008. The last three digits are 008, which is divisible by 8 (008 / 8 = 1). Thus, 1008 is divisible by 8.

Step 3: Combine the Results

If a number passes both the divisibility tests for 3 and 8, it's divisible by 24.

Example 1: Is 792 divisible by 24?

1. Divisibility by 3: 7 + 9 + 2 = 18. 18 is divisible by 3. (Pass)
2. Divisibility by 8: The last three digits are 792. 792 / 8 = 99. (Pass)

Conclusion: Since 792 passes both tests, it is divisible by 24.

Example 2: Is 1008 divisible by 24?

1. Divisibility by 3: 1 + 0 + 0 + 8 = 9. 9 is divisible by 3. (Pass)
2. Divisibility by 8: The last three digits are 008, which is divisible by 8. (Pass)

Conclusion: Since 1008 passes both tests, it is divisible by 24.

Example 3: Is 252 divisible by 24?

1. Divisibility by 3: 2 + 5 + 2 = 9. 9 is divisible by 3. (Pass)
2. Divisibility by 8: The last three digits are 252 which is NOT divisible by 8 (252/8 = 31.5). (Fail)

Conclusion: Because it fails the divisibility by 8 test, 252 is NOT divisible by 24.

Why This Method Works

This method works because of fundamental mathematical principles. Since 24 = 8 x 3, a number divisible by 24 must be divisible by both 8 and 3. The individual divisibility rules for 8 and 3 provide efficient ways to check this.

Practice Makes Perfect

The best way to master this divisibility rule is to practice. Try applying these steps to various numbers. You'll quickly become proficient at determining divisibility by 24. This skill is valuable in various mathematical contexts, from simplifying fractions to solving more complex problems.