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### Course: Class 6 (Old)>Unit 3

Lesson 3: Tests for divisibility

# The why of the 9 divisibility rule

Why you can test divisibility by 9 by just adding the digits. Created by Sal Khan.

## Want to join the conversation?

• Maybe I missed this...can you do the same trick where you add the numbers for six since it is divisable by three? Three and nine work. Thanks for any anwsers!
• It doesn’t work for 6 the same way as 3 and 9. If it did, you could say,
18: 1+8 = 9, which is not divisible by 6 even though 18 is. So it doesn't work.

However, as one person suggested but didn’t complete, you can see that if the number were divisible by 2 and 3 then that would make the number divisible by 6. So if the number ends in an even number (0,2,4,6,8) and the digits sum to a number divisible by 3, then the original number is divisible by 6. So for 18:
It ends in 8, which is even, so 18 is divisible by 2.
1+8=9, which is divisible by 3, so 18 is divisible by 3.
Since the number 18 is divisible by both 2 and 3, then it is divisible by 6.
• When figuring out if a number is divisible by 9. Can I just cross out the 9's, cross out numbers that will equal 9 when added (8+1, 7+2, 6+3, 5+4). Then just add the rest? Example: Is 697,225,842 divisible by 9? First crossing out 9's, then the 7 and a 2 because adding them would equal 9. Then doing the same with the 4 and 5. So now I have 6 _ _ , _ 2 _ , 8 _ 2.... Adding these give me 18. YES this is divisible by 9. Can this be a more efficient way to figure out numbers that are divisible by 9?
• You are just ignoring part of the number that you already know is divisible by 9, and then seeing of the remainder is divisible by 9. The whole process is like a type of borrowing. For example, 18 is divisible by 9 because 18 is a 10 and an 8. If the 8 borrows 1 from 10 then they are both 9. So 18 is two 9s. If you have 918, then you have 100 9s and 2 9s, for a total of 102 9s. You can ignore either the 900 or the 18 because they are each divisible by 9.
• why did you think of this and how
• why is it always (for example) 1+99... why is it 99? sure, 1+99=100, but so is 52+48+100
• And because 1 times a number is itself. We want to reach a point where the sum of the digits is isolated and the rest of the numbers are divisible by 9.
(1 vote)
• Is six also able to have this rule?
• No for 6 we have to see whether the number is divisible by its factors(2,3) or not.
• I don't understand the distributive form he's doing
• Well, pretend you have 3*(3+5). That is the same thing as3*3+3*5. It's 9+15=24, same as 3+5 (8)*3 (24)
• How i can know how much divisors have a number?? For example the 24 have 8 divisors: 1,2,3,4,6,8,12,24 but how i know it more quickly?
• That's a tough one, trial and error (might want to begin at 2 and work your way up) seems to be the only method. Maybe you could be the next mathematician to invent a revolutionary new way to quickly find the divisors of a number. Best of luck :)
• this trick works only for 3, 6, 9, and on by 3's, right? things divisible by 3. also, can someone explain how to 9's trick for multiplication tables on your fingers works? I
think it's something to do with the fact that if you put one of ten fingers down you always have 9 left.