Properties of divisible numbers

Question

Here is a list of divisible number properties.

To find if number x is divisible by: 
 2 : If x is even, True
 3 : If the sum of the digits of x are a multiple of 3, True
 4 : If the ones and tens digits form a number that is divisible by 4, True
 5 : If the ones digit is a 0 or 5, True
 6 : If x is divisible by 2 AND 3, True
 7 : Double the ones digit and subtract the last number from the remaining digits. If difference divisible by 7, True
 8 : If last 3 digits form a number that is divisible by 8, True  OR  If x is divisible by 2 three times, True
 9 : If the sum of the digits are a multiple of 9, True
 10 : If the ones digit is 0, True
 11 (Method 1) : Add each digit using these properties: - + - +... If the resulting number is divisible by 11, True
 11 (Method 2) : Starting with ones digit, add every other number (A). Add the remaining numbers (B). If A - B is divisible by 11, True
 12 : If sum of the digits is a multiple of 3 and the last two digits are a multiple of 4, True
 15 : If x is divisible by 3 AND 5, True

4 Example:  Is 312 divisible by 4?
If the ones and tens digits form a number that is divisible by 4 then true. The ones and tens digits form 12 and 12 is divisible by 4, therefore true.

7 Example:  Is 357 divisible by 7?
Double the ones digit (7) to get 14. Subtract 14 from remaining digits (35) to get 21. 21 is divisible by 7, therefore true.

9 Example:  Is 95,301 divisible by 9?
The number 95,301 is divisible by 9 because the digits add to 18 (9+5+3+0+1), which is a multiple of 9.

11 (Method 1) Example:  Is 824,472 divisible by 11?
-8 + 2 - 4 + 4 - 7 + 2 = -11, which is divisible by 11, therefore 824,472 is divisible by 11.

11 (Method 2) Example:  Is 824,472 divisible by 11? 
Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.

If anyone knows any other divisible number properties, please list them. Thanks.

timetrader · Answer

Great initiative!

You might be interested in these two posts from Basic Mathematical principles
basic-mathematical-principles-14576.html?view-post=90425#p90425
basic-mathematical-principles-14576.html?view-post=154753#p154753

Keep it up! I would love to see a compilation on divisibility rules!

I3igDmsu · Answer

Thanks, I added a few more.

IanStewart · Answer

There's no reason to exclude 0. Zero is definitely divisible by 2;

You mean: if the tens and ones digits form a number which is divisible by 4. The number 31,212 is divisible by 4, for example, because the last two digits form a number (12) which is divisible by 4, even though the tens digit and the ones digit are not individually divisible by 4.

There's no reason to exclude 0. Zero is definitely divisible by 5;

Again for clarity - the last three digits should form a number divisible by 8. For example, 85,328 is divisible by 8 because 328 is divisible by 8 (328 = 8*41).

You mean: If the sum of the digits is a multiple of 9. The sum does not need to equal 9. The number 95,301 is divisible by 9, for example, because the digits add to 18, which is a multiple of 9.

There's no reason to exclude 0. Zero is definitely divisible by 10;

I3igDmsu · Answer

IanStewart - great comments. I've updated everything above.

bb · Answer

Thanks! Kudos Added to the Math Tips Thread: new-to-the-math-forum-please-read-this-first-77764.html

eddiebeherano · Answer

Ok there's maybe one little simplification for the divisibility of number 11:

There's one scheme applied to the digits : - + - + etc. if the forming number is divisible by 11 then our number is divisible by 11.

Lets take for ex 3916 which is divisible by 11.
Application : -3 + 9 - 1 + 6 = 11 -----> so 3916 is divisible by 11.

Another example. 1099989

-1+0-9+9-9+8-9 = -11 --------. so 1099989 is divisible by 11.

Hope it helps.

Eddie

anhnguyen990 · Answer

Thanks ! It helps a lot

BeeSkool · Answer

7 : Double the ones digit and subtract the last number from the remaining digits. If difference divisible by 7, True I cannot for the life of me figure out how this works. :?:

Would anybody be kind enough to break this methodology down with a specific example so I can see what I'm doing wrong? Thank you!

udini · Answer

@BeeSkool
Consider the number 273 , 
if you want to apply the rule 
 
then you have to 
- double the ones digit : 2 x 3 =6
- subtract 6 from the number formed by the remaing digits ( = 27) : 27 -6 = 21

21 is divisible by 7 => 273 is divisible too

BeeSkool · Answer

Awesome thanks! :-D

rsvaishu · Answer

amazing for 7 and 11 properties and thanks for sharing it

Robiou · Answer

Bah my post got deleted. Thanks for the table. By the way, on page 244 Kaplan Premier they—incorrectly—state that, "...you can combine these rules above with factorization tom figure out whether a number is divisible by other numbers." the example given is that 8184 is divisible by 44 because it is divisible by (4 and 11). While this may be true in this case, it should advise that you use prime factorization.

For example, 36 is divisible by both 4 and 2, yet it is not divisible by 8. Rather, it should be divisible by 2 three times.

Posted from GMAT ToolKit

For example, 36 is divisible by both 4 and 2, yet it is not divisible by 8. Rather, it should be divisible by 2 three times.

https://gmatclub.com/static/posted_from.png     Posted from    GMAT ToolKit

laurapv · Answer

Awesome, the MGMAT Number Properties didn't list the properties for 7,11, 12 and 15.

PadawanOfTheGMAT · Answer

Great addition to MGMAT Number Properties.

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Properties of divisible numbers

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