How to Factor Using Divisibility Rules

Divisibility rules make factorization simpler.

Finding the divisors of a number is necessary to determine its prime factors, or the least common multiple or greatest common factor of two numbers. However, finding the divisors of a number can become a time-consuming process, and knowing the divisibility rules makes the task quicker and easier. The divisibility rules tell us which number is exactly divisible by another. These rules are most useful when the number being factorized is large.

Things You'll Need

• Calculator (optional)

Instructions

1. Check if the number is divisible by 2. All numbers ending with zero or an even number are divisible by 2. Suppose it is required to find the factors of 422. Since it ends with an even number, it is divisible by 2.

2. Write the number as a product of its factors as you find them. In this case, 422 becomes 2 * 211. The task now is to find the factors of the remaining largest factor, 211. 211 is not divisible by 2, so continue to try out greater possible factors.

3. Determine if the remaining factor is divisible by 3 by taking the sum of the digits. If the sum is divisible by 3, then so is the number. The digits in 211 add up to 4. Since 4 is not divisible by 3, the number 211 is also not divisible by 3.

4. Take the last two digits of the number and check if the number they form is divisible by 4. Since 11 is not divisible by 4, the number 211 is also not divisible by 4.

5. Remember that for any number to be divisible by 5, it must end in 0 or 5. Using this rule, 211 is not divisible by 5.

6. Check if the number is divisible by 6. Any number divisible by both 2 and 3 is also divisible by 6.

7. Apply the divisibility rule for 7 by taking the last digit, doubling it, and subtracting it from the rest of the number. In this example, double of one is two, and 21 minus two is 19, which is not divisible by seven. Therefore, the number 211 is also not divisible by seven.

8. Take the last three digits of a number and check if the number they form is divisible by 8. Therefore 211 is not divisible by eight.

9. Add the digits together to apply the divisibility rule for the number 9. If the sum is divisible by 9, so is the number. The sum of the digits of 211 is 4, which is not divisible by 9.

10. Note that for a number to be divisible by 10, it must end with a zero.

11. Start with zero and add the leftmost digit of the number. Subtract the next digit, and continue alternately to add and subtract to apply the divisibility rule for 11. For 211, two minus one plus one equals two, which is not divisible by 11.

12. Remember that any number divisible by both 3 and 4 is also divisible by 12.

13. Subtract nine times the last digit of the number from the remaining number. If the result is divisible by 13, then the number is also divisible by 13. In this case: nine times one is nine and subtracting nine from 21 gives 12, which is not divisible by 13. Further divisions by numbers up to the square root of 211 show that it is a prime number. Therefore, the factors of 422 are 2 and 211.

Tips & Warnings

• A number is shown to be a prime if it is not divisible by any numbers up to its square root.

• The divisibility rules tend to get more complex with larger divisors. For divisors greater than 13, use a calculator to check whether a number divides into another.

References

• Wolfram MathWorld; "Fundamental Theorem of Arithmetic"; Eric W. Weisstein
• Purplemath; "Factoring Numbers"; Elizabeth Stapel
• Math Forum: Divisibility Rules