Divisibility Rules for Prime Numbers: A Comprehensive Guide

Summary

Divisibility rules offer shortcuts to test if a number is divisible by another without performing division. This article explores these rules for prime numbers below 50, providing clear examples and a detailed proof for the rule of 7. Learn how to apply these rules effectively and understand the underlying principles.

Introduction

Divisibility rules are essential tools in mathematics, allowing us to determine if a number is divisible by another without performing the actual division. These rules simplify the process by transforming a given number's divisibility by a divisor into a smaller number's divisibility by the same divisor. If the result is not immediately obvious, the rule can be reapplied to the smaller number.

Common Divisibility Rules

In children's math textbooks, you will often find divisibility rules for numbers like 2, 3, 4, 5, 6, 8, 9, and 11. However, rules for prime numbers such as 7 are rarely included. This article aims to fill that gap by presenting divisibility rules for prime numbers below 50.

General Divisibility Rule for Prime Numbers

To find the divisibility rule for any prime number ( p ):

1. Identify the smallest multiple of ( p ) such that adding 1 results in a multiple of 10.
2. Divide this multiple by 10 to get a natural number ( n ).
3. Calculate ( p - n ).

Example Calculations

For Prime Number 7

• Multiples of 7: ( 1 \times 7, 2 \times 7, 3 \times 7, 4 \times 7, 5 \times 7, 6 \times 7, 7 \times 7 )
• ( 7 \times 7 = 49 ) and ( 49 + 1 = 50 ) (a multiple of 10)
• ( n = \frac{50}{10} = 5 )
• ( p - n = 7 - 5 = 2 )

For Prime Number 13

• Multiples of 13: ( 1 \times 13, 2 \times 13, 3 \times 13 )
• ( 3 \times 13 = 39 ) and ( 39 + 1 = 40 ) (a multiple of 10)
• ( n = \frac{40}{10} = 4 )
• ( p - n = 13 - 4 = 9 )

Table of Values for Primes Below 50

Applying the Divisibility Rule

To determine if a number is divisible by ( p ):

1. Take the last digit of the number.
2. Multiply it by ( n ) and add it to the rest of the number, or multiply it by ( p - n ) and subtract it from the rest of the number.
3. If the result is divisible by ( p ) (including zero), then the original number is divisible by ( p ).
4. If the new number's divisibility is unclear, reapply the rule.

Divisibility Rule for 7

For 7, ( p - n = 2 ) is lower than ( n = 5 ).

Rule: To check if a number is divisible by 7, take the last digit, multiply it by 2, and subtract it from the rest of the number. If the result is divisible by 7, then the original number is divisible by 7.

Example

Check if 49875 is divisible by 7:

1. Twice the last digit: ( 2 \times 5 = 10 )
2. Rest of the number: 4987
3. Subtract: ( 4987 - 10 = 4977 )
4. Repeat: ( 2 \times 7 = 14 ), ( 497 - 14 = 483 )
5. Repeat: ( 2 \times 3 = 6 ), ( 48 - 6 = 42 )
6. 42 is divisible by 7, so 49875 is divisible by 7.

Divisibility Rule for 13

For 13, ( n = 4 ) is lower than ( p - n = 9 ).

Rule: To check if a number is divisible by 13, take the last digit, multiply it by 4, and add it to the rest of the number. If the result is divisible by 13, then the original number is divisible by 13.

Example

Check if 46371 is divisible by 13:

1. Four times the last digit: ( 4 \times 1 = 4 )
2. Rest of the number: 4637
3. Add: ( 4637 + 4 = 4641 )
4. Repeat: ( 4 \times 1 = 4 ), ( 464 + 4 = 468 )
5. Repeat: ( 4 \times 8 = 32 ), ( 46 + 32 = 78 )
6. 78 is divisible by 13, so 46371 is divisible by 13.

Proof of Divisibility Rule for 7

Proof

Let ( D ) (greater than 10) be the dividend. Let ( D1 ) be the unit's digit and ( D2 ) be the rest of the number of ( D ).

[ D = D1 + 10D2 ]

We need to prove:

1. If ( D2 - 2D1 ) is divisible by 7, then ( D ) is also divisible by 7.
2. If ( D ) is divisible by 7, then ( D2 - 2D1 ) is also divisible by 7.

Proof of (i)

If ( D2 - 2D1 ) is divisible by 7, then:

[ D2 - 2D1 = 7k ]

Multiplying both sides by 10:

[ 10D2 - 20D1 = 70k ]

Adding ( D1 ) to both sides:

[ (10D2 + D1) - 20D1 = 70k + D1 ]

[ 10D2 + D1 = 70k + 21D1 ]

[ D = 70k + 21D1 = 7(10k + 3D1) ]

Thus, ( D ) is divisible by 7.

Proof of (ii)

If ( D ) is divisible by 7, then:

[ D1 + 10D2 = 7k ]

Subtracting ( 21D1 ) from both sides:

[ 10D2 - 20D1 = 7k - 21D1 ]

[ 10(D2 - 2D1) = 7(k - 3D1) ]

Since 10 is not divisible by 7, ( D2 - 2D1 ) must be divisible by 7.

Conclusion

Divisibility rules for prime numbers provide a quick and efficient way to determine if a number is divisible by a prime without performing division. By understanding and applying these rules, you can simplify many mathematical problems. The method of finding ( n ) can be extended to prime numbers above 50 as well.

For more information on divisibility rules, visit Math is Fun.

This article has provided a comprehensive guide to understanding and applying divisibility rules for prime numbers below 50. By following the steps and examples provided, you can easily determine the divisibility of numbers by these primes.