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how to check if a number is divisible by 75

how to check if a number is divisible by 75

2 min read 21-01-2025
how to check if a number is divisible by 75

Knowing divisibility rules is a handy skill for mental math and simplifying calculations. This article will show you how to quickly determine if a number is divisible by 75 without using a calculator. We'll explore the simple method and provide examples to solidify your understanding.

Understanding Divisibility by 75

The key to understanding divisibility by 75 lies in its prime factorization. 75 can be broken down into 3 x 5 x 5, or 3 x 5². This means a number is divisible by 75 if it's divisible by both 3 and 25 (since 25 = 5 x 5). We'll use this fact to create our divisibility test.

The Two-Step Divisibility Test for 75

To check if a number is divisible by 75, follow these two simple steps:

  1. Divisibility by 3: Check if the sum of the digits is divisible by 3. If the sum of a number's digits is divisible by 3, then the number itself is divisible by 3.

  2. Divisibility by 25: Check if the last two digits of the number are divisible by 25. The numbers divisible by 25 end in 00, 25, 50, or 75.

If both conditions are met (divisible by 3 AND divisible by 25), then the original number is divisible by 75. If either condition fails, the number is not divisible by 75.

Examples: Checking for Divisibility by 75

Let's work through a few examples to illustrate the process:

Example 1: Is 750 divisible by 75?

  1. Divisibility by 3: 7 + 5 + 0 = 12. 12 is divisible by 3 (12 / 3 = 4).

  2. Divisibility by 25: The last two digits are 50, which is divisible by 25 (50 / 25 = 2).

Result: Since both conditions are true, 750 is divisible by 75 (750 / 75 = 10).

Example 2: Is 1575 divisible by 75?

  1. Divisibility by 3: 1 + 5 + 7 + 5 = 18. 18 is divisible by 3 (18 / 3 = 6).

  2. Divisibility by 25: The last two digits are 75, which is divisible by 25 (75 / 25 = 3).

Result: Both conditions are met, therefore 1575 is divisible by 75 (1575 / 75 = 21).

Example 3: Is 2250 divisible by 75?

  1. Divisibility by 3: 2 + 2 + 5 + 0 = 9. 9 is divisible by 3 (9 / 3 = 3).

  2. Divisibility by 25: The last two digits are 50, which is divisible by 25 (50 / 25 = 2).

Result: Both conditions are true, so 2250 is divisible by 75 (2250 / 75 = 30).

Example 4: Is 1050 divisible by 75?

  1. Divisibility by 3: 1 + 0 + 5 + 0 = 6. 6 is divisible by 3 (6/3 = 2).

  2. Divisibility by 25: The last two digits are 50 which is divisible by 25 (50/25 = 2).

Result: Both conditions are met, therefore 1050 is divisible by 75 (1050/75 = 14).

Example 5: Is 1234 divisible by 75?

  1. Divisibility by 3: 1 + 2 + 3 + 4 = 10. 10 is NOT divisible by 3.

Result: Because the first condition fails, 1234 is NOT divisible by 75.

Conclusion

By using the combined divisibility rules for 3 and 25, determining if a number is divisible by 75 becomes a straightforward process. Practice these steps with different numbers, and you'll quickly master this useful divisibility check! Remember, if a number isn't divisible by both 3 and 25, it's not divisible by 75.

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