Divisibility Test for 17: Easy Methods and Techniques

Divisibility rules are a fundamental concept in number theory, helping to quickly determine if a number can be divided by another without leaving a remainder. Among the numerous divisibility tests, the test for 17 stands out for its unique properties and various methods. This article explores the most effective methods to check if a given number is divisible by 17, including step-by-step examples.

The Double and Subtract Method

The double and subtract method is a simple yet effective technique to test if a number is divisible by 17. Here's how it works:

Double the last digit of the number. Subtract this doubled value from the rest of the number (excluding the last digit). If the result is divisible by 17 (including 0), then the original number is also divisible by 17. Otherwise, it is not. For numbers that yield a large result, repeat the process until a manageable number is achieved.

Example: Let's test if 153 is divisible by 17.

Double the last digit: 3 × 2 6. Subtract this doubled value from the rest of the number: 15 - 6 9. Since 9 is not divisible by 17, 153 is not divisible by 17.

Grouping Method and Multiplication Technique

A more advanced method involves grouping the digits of the number and performing a series of multiplications and subtractions. This technique helps in simplifying complex numbers into manageable units. Here’s how it works:

Take the last two or three digits and multiply them by 9. Subtract this product from the rest of the number. Repeat the process until a double-digit number is reached. If the final result is 17, the original number is divisible by 17.

Example: Test if 210065413 is divisible by 17.

Group 2100654 - 13. Multiply 13 by 9: 13 × 9 117. Subtract this product from the remaining number: 2100654 - 117 2100537. Repeat: 21005 - 9 × 37 20672; 206 - 9 × 72 -442; 4 - 9 × 42 -374. Check if -374 is divisible by 17: -374 ÷ 17 -22.

Since -374 is divisible by 17, the original number 210065413 is also divisible by 17.

Multiplication and Addition Method

This method involves a combination of multiplication and addition to achieve the divisibility test for 17:

Multiply the last digit by 12 and add it to the rest of the number. Continue this process until you reach a double-digit number. If the final result is 17, the original number is divisible by 17.

Example: Test if 2601 is divisible by 17:

Start with 2601, double the last digit (1) and add it to 260: 2601 12 272. Repeat: 272 212 51. Repeat again: 51 12 512. Repeat one last time: 512 12 17.

Since 17 is divisible by 17, 2601 is also divisible by 17, and the quotient is 253.

Subtracting a Multiple of the Last Digit

This method is based on the concept of relative primality and involves a subtraction step:

Subtract 5 times the last digit from the rest of the number. Check if the result is divisible by 17. If it is, the original number is also divisible by 17.

Example: Let's test if 1045 is divisible by 17 using this method:

Multiply the last digit (5) by 5: 5 × 5 25. Subtract this product from the rest of the number: 104 - 25 79. 79 is not divisible by 17, so 1045 is not divisible by 17.

Conclusion

The divisibility test for 17 has several methods, each with its nuances. While some techniques are straightforward, others require more steps but can be more efficient for larger numbers. Understanding these methods not only helps in verifying divisibility but also deepens one's knowledge of number theory.

By exploring and practicing these techniques, you can quickly determine the divisibility of any number by 17, enhancing your skills in basic arithmetic and number theory.