Divisibility Rules

Divisibility is a property indicating that a number can be entirely divided by another number; i.e., there will not be a remainder.

|54\div 6=9 \text{ remainder}\ 0|, thus |54| is divisible by |6|.

|22\div 5=4 \text{ remainder}\ 2|, thus |22| is not divisible by |5|.

Divisibility rule for division by |2|

1. Identify the number in the ones position.

2. Check if it is even.

3. If the number is even, then the number is divisible by |2|.

Is the number |10\ 256| divisible by |2|?

1. The digit in the ones position is |6|.

2. |6| is an even number.

3. Thus, |10\ 256| is divisible by |2|.

Divisibility rule for division by |3|

1. Add all the digits in the number.

2. Check if this sum is divisible by |3|.

3. If the sum is divisible by |3|, then the number is also divisible by |3|.

Is the number |261| divisible by |3|?

1. |2+6+1=\color{red}{9}|

2. |\color{red}{9}| is divisible by |3|. |\left(9\div 3=3\right)|

3. Thus, |261| is divisible by |3|.

Divisibility rule for division by |4|

1. Identify the number formed by the last two digits (tens and ones).

2. Check if this number is divisible by |4| or if it ends in |00|.

3. If the number formed by its last two digits is divisible by |4| or if it ends in |00|, then the original number is divisible by |4|.

Is the number |12\ 524| divisible by |4|?

1. The number formed by the last two digits of |12\ 524| is |\color{red}{24}|.

2. |\color{red}{24}| is divisible by |4|. |\left(24\div 4=6\right)|

3. Thus, |12\ 524| is divisible by |4|.

Divisibility rule for division by |5|

1. Identify the number in the ones position.

2. If the ones digit is |0| or |5| , then the number is divisible by |5|.

Is the number |325\ 465| divisible by |5|?

1. The digit in the ones position is |\color{red}{5}|.

2. Since the ones digit is |\color{red}{5}|, |325\ 465| is divisible by |5|.

Divisibility rule for division by |6|

1. Check if the number is divisible by |2| (is the number even?)

2. Check if the number is divisible by |3| (is the sum of the digits divisible by |3|?)

3. If the number is divisible by both |2| and |3|, then the number is divisible by |6|.

Is the number |5\ 364| divisible by |6|?

1. The ones digit is even |\left(\color{red}{4}\right)|, so the number is divisible |2|.

2. By adding all the digits of the number, we get |5+3+6+4=\color{red}{18}|. |\color{red}{18}| is divisible by |3|, so the number is divisible by |3|.

3. Since |5\ 364| is divisible by both |2| and |3|, it is divisible by |6|.

Divisibility rule for division by |8|

1. Identify the number formed by the last three digits (hundreds, tens, and ones).

2. Check if this number is divisible by |8|.

3. If this number is divisible by |8|, then the original number is also divisible by |8|.

Is the number |10\ 168| divisible by |8|?

1. The number formed by the last three digits is |168|.

2. |168| is divisible by |8|. |\left(168\div 8=21\right)|

3. Thus, |10\ 168| is divisible by |8|.

Divisibility rule for division by |9|

1. Add all the digits in the number.

2. Check to see if the sum is divisible by |9|.

3. If the sum is divisible by |9|, then the original number is divisible by |9|.

Is the number |3\ 159| divisible by |9|?

1. |3+1+5+9=\color{red}{18}|

2. |\color{red}{18}| is divisible by |9|. |\left(18\div 9=2\right)|

3. Thus, |3\ 159| is divisible by |9|.

Divisibility rule for |10|.

1. Identify the last digit in the number (the ones position).

2. If the number is |0|, then the number is divisible by |10|.

Is the number |125\ 890| divisible by |10|?

1. The last digit is |0|.

2. Since the number is |0|, |125\ 890| is divisible by |10|.

Divisibility rule for |12|

1. Check if the number is divisible by |3.|

2. Check if the number is divisible by |4.|

3. If the number is divisible by both |3| and |4,| then the number is divisible by |12.|

Is the number |216| divisible by |12|?

1. When we add all the digits in the number, we obtain |2+1+6=\color{red}{9}.|
   |\color{red}{9}| is divisible by |3,| thus |216| is also divisible by |3.|

2. The number formed by the last two digits is |16.| Since this number is divisible by |4,| then |216| is also divisible by |4.|

3. Thus, |216| is divisible by |12.|

Divisibility rule for |25|

1. Identify the last two digits of the original number (tens and ones).

2. If these last two digits are |00|, |25|, |50|, or |75|, then the original number is divisible by |25|.

Is |2\ 575| divisible by |25|?

1. The last two digits of |2\ 575| are |75|.

2. Since the last two digits are |75|, |2\ 575| is divisible by |25|.

More divisibility rules:

• Divisibility rule for |7|:

  A number is divisible by |7| if its number of tens minus twice the number in the ones position is divisible by |7.|

Is the number |294| divisible by |7|?

Number of tens: |29|
Ones digit: |4|

|29 - (2\times 4) = 21|

|21| is divisible by |7|. |\left(21\div7=3\right)|

Thus, |294| is divisible by |7|.

More divisibility rules:

• Divisibility rule for |11|:

  A number is divisible by |11| if the difference between the sum of the digits located in odd positions (ones, hundreds, etc.) and the sum of the digits located in even positions (tens, thousands, etc.) is divisible by |11|:

Example 1:
Is the number |495| divisible by |11|?

Sum of the digits in odd positions: |4+5=9|
Sum of the digits in even positions: |9=9|
Difference between the two sums: |9-9=0|

Since |0| is divisible by any number (in this case by |11| ), |495| is also divisible by |11.|

Example 2:
Is the number |10\ 989| divisible by |11|?

Sum of the digits in odd positions: |1+9+9=19|
Sum of the digits in even positions: |0+8=8|
Difference between the two sums: |19-8=11|

Since |11| is divisible by |11|, |10\ 989| is also divisible by |11|.

More divisibility rules:

• Divisibility rule for |13|:

  A number is divisible by |13| if the number of tens plus four times the number in the ones position is divisible by |13.|

Is the number |117| divisible by |13|?

Number of tens: |11| 
Ones digit: |7|  

|11 + (4\times 7) = 39|  

|39| is divisible by |13|. |\left(39\div13=3\right)| 

Thus, |117| is divisible by |13|.

More divisibility rules:

• Divisibility rule for |14|:
  A number is divisible by |14| if it is divisible by both |2| and |7.|

• Divisibility rule for |15|:
  A number is divisible by |15| if it is divisible by both |3| and |5.|

• Divisibility rule for |20|:
  A number is divisible by |20| if the number formed by its last two digits (tens and ones) is divisible by |20.|

• Divisibility rule for |100|:
  A number is divisible by |100| if it ends with at least two |00.|