Rules of divisibility

It can sometimes be useful to be able to quickly figure out what numbers a given number can be divided by. For example, you may be trying to simplify a fraction or a radical expression. In these situations, it’s a good idea to remember the rules of divisibility.

Rules of divisibility

• 2: The ones digit is 2, 4, 6, 8, or 0.
• 3: Add the digits in the number together. If the resulting number is divisible by 3, then the original number is divisible by 3.
• 4: If the last two digits in the number are divisible by 4, then the original number is divisible by 4.
• 5: The ones digit is 5 or 0.
• 6: If the number is divisible by both 2 and 3, then it is divisible by 6.
• 8: If the last three digits are divisible by 8, then the number is divisible by 8.
• 9: Add the digits in the number together. If the resulting number is divisible by 9, then the original number is divisible by 9.
• 10: The ones digit is 0.

There have been several recent attempts to create a simple rule for checking for divisibility by 7, but so far nothing has proven simple enough to include here.

To check if a number is divisible by one of the numbers on this list, simply run it through the test. For example,  if you want to know if 115 is divisible by 3, add the digits together. In this case, 1 + 1 + 5 = 7. 7 is not divisible by 3, so 115 is not divisible by 3.

Try taking a number and testing it for each rule.  I’ll demonstrate using 249. Let’s see what it can be easily divided by.

• 2: The last digit of 249 is 9. 9 is not 2, 4, 6, 8, or 0, so 249 is not divisible by 2.
• 3: 2 + 4 + 9 = 15. 15 is divisible by 3, so 249 is divisible by 3.
• 4: 49 is not evenly divisible by 4, so 249 is not divisible by 4.
• 5: The last digit of 249 is 9. 9 is not 5 or 0, so 249 is not divisible by 5.
• 6: We have already proven that 249 is not divisible by 2, so it is also not divisible by 6.
• 8: 249 divided by 8 gives a remainder, so it is not divisible by 8.
• 9: 2 + 4 + 9 = 15. 15 is not divisible by 9, therefore 249 is not divisible by 9.
• 10: The last digit of 249 is 9. 9 is not 0, so 249 is not divisible by 10.

According to our work, 249 is divisible only by 3.

Once you get the hang of it, checking the rules of divisibility goes pretty quickly.