Determining Divisibility by 27: Methods and Tips

Divisibility is a fundamental concept in number theory, and understanding how to determine if a number is divisible by 27 can be particularly useful in various mathematical and practical applications. This article explores different methods to check for divisibility by 27, including the sum of digits, direct division, and specialized techniques.

Why 27?

27 is a unique number because it is a power of 3, specifically 33. This characteristic influences its divisibility rules and provides distinct methods for checking whether a number is divisible by 27.

Checking for Divisibility by 27

Direct Division Method

The simplest and most direct method to determine if a number is divisible by 27 is to perform the division directly. If the quotient is an integer (i.e., no remainder), then the number is divisible by 27.

Example: Let's check if 135 is divisible by 27.

Perform the division: 135 ÷ 27 5

Since the answer is an integer, 135 is divisible by 27.

Divisibility by 9 and 3

Since 27 is 33, a number is divisible by 27 if and only if it is divisible by both 3 and 9. This leads to the following steps:

Check if the number is divisible by 3 by summing its digits. If the sum is a multiple of 3, proceed to the next step.

Check if the sum of the digits (as obtained in the previous step) is divisible by 9.

Multiplying the Last Digit by 8

Another method involves taking the last digit of the number, multiplying it by 8, and subtracting the result from the rest of the number. If the final result is divisible by 27, then the original number is also divisible by 27.

Example: Check if 135 is divisible by 27.

Multiply the last digit (5) by 8: 5 × 8 40

Subtract this result from the rest of the number: 13 - 40 -27

Since -27 is divisible by 27, 135 is also divisible by 27.

Special Techniques for Large Numbers

For very large numbers, there is a specialized technique that uses the fact that 273 999. If a number is greater than 1000, you can add every 3 digits together and check if the sum is divisible by 27.

Example: Check if 123456 is divisible by 27.

Add every 3 digits: (123 456 000 579)

Check if 579 is divisible by 27. Since 579 ÷ 27 21.4444, it is not divisible by 27.

Factors of 27

Another way to determine if a number is divisible by 27 is to check if the number itself is a multiple of 27, which means one of its factors must be 27.

Conclusion

In conclusion, while there isn't a simple trick for checking divisibility by 27 like there is for smaller numbers, there are several reliable methods. The most straightforward is the direct division method, followed by the technique involving the sum of digits and the specialized method for large numbers.

Understanding these methods can help you quickly determine if a number is divisible by 27, which is particularly useful in mathematical computations and problem-solving scenarios.