DIVISIBILITY

When solving complex problems, we can encounter a situation where we want to know if a large number is divisible by our regular digits.

In higher level courses in this site, we may use actual proofs to test for divisibility.

  

Divisibility by 2:

An even integer is divisible by 2. Even numbers are numbers that have a last digit of 0, 2, 4, 6, 8.

 

Divisibility by 3:

Integers whose digits add up to a multiple of 3 (like adding up to 6, 9, 12 and so on…) are divisible by 3.

. This number is divisible by 3.

 

Divisibility by 4:

If the last two digits of an integer form a number that is divisible by 4, then that integer is divisible by 4.

145696: the two last digits form 96. 96/4=24. So the number 145696 is divisible by 4.

 

Divisibility by 5:

An integer whose last digit is 0 or 5 is divisible by 5.

: The last digit is 5. So 3455 is divisible by 5.

Same goes for .

 

Divisibility by 6:

Integers whose digits add up to a multiple of 3 and are even (like adding up to 36, 72, 12 and so on…) are divisible by 6.

. This number is divisible by 6.

 

Divisibility by 7:

To find out if an integer  is divisible by 7, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don’t know the new number’s divisibility, you can apply the rule again.

:

(Divisible by 7)

More challenging:

(Still don’t know)

 

(Divisible)

 

Divisibility by 8

An integer, whose last 3 digits form a number that is divisible by 8, is divisible by 8.

. Here the number is then divisible by 8.

 

Divisibility by 9:

An integer is divisible by 9 if its digits add up to a multiple of 9.

. Also . Finally, the number 51984 is divisible by 9.

 

Divisibility by 10:

An integer is divisible by 10 if the last digit is 0

:  The last digit is 0. is divisible by 10.

 

Divisibility by 11:

To check the divisibility by 11, we should start from the right and add every second digit. Let’s call that sum S1.

Then add the remaining digits. Let’s call this new sum S2

If S1-S2 is divisible by 11, then the given number is divisible by 11.

(Not divisible by 11)

 

Now:

(Divisible by 11)