Rules, rules, rules -- No one likes to follow rules, but sometimes there are good rules, like the ones that allow you to do less work! Well, these are exactly the kind of rules that you find on this page. There are many shortcuts or tricks that allow you to test whether a number, or dividend, is divisible by a given divisor. This page focuses on the most-frequently studied divisibility rules which involve divisibility by 2,4,5,6,8,9, 10 and 11.

### A word of caution about hand held calculators

Keep this in mind before you buy graphing calculator: Even many of the best graphing calculators or this free online graphing calculator lose their accuracy when they start dividing large numbers such as 12,347,496,132. Try to divide that large number by 11. If your calculator outputs that 12,347,496,132 is divisible by 11, your calculator IS WRONG (Look at the last example at the bottom for details). When you're dealing with exceedingly large numbers, you should rely, whenever possible, on the rules on this page rather than a scientific or a graphing calculator.

So, you may be asking yourself why is this page's calculator more accurate than that graphing calculator you bought?

- The answer is : the calculators on this page use the rules for divisibility to calculate their answers (instead of the division algorithm). Your hand held calculator uses algorithms that work for all numbers and, if you're really interested, in learning more about why computers (not just calculators) do not accurately divide huge numbers look up floating point error, sometimes called rounding error.

### Divisibility Calculator

If you do not have the time or inclination to use the rules below, you can check to see if two numbers are divisible by using the divisibility calculator below. Just fill in the numbers and let the divisibility calculator do the rest.

### Divisibility by 2 Rule

Almost everyone is familiar with this rule which states that any even number can be divided by 2. Even numbers are multiples of 2. A number is even if ends in 0,2,4,6, or 8.

Use the divisibility calculator below to determine if any number is divisible by two. Type in any number that you want, and the calculator will use the rule for divisibility by 2 to explain the result.

See what the rule for divisibility by two has to say about the following number:

**Examples** of numbers that **are even ** and therefore pass this divisibility test

- 2
- 0
- 4
- -2
- -312
- 31,102

**Examples** of numbers that are **do not pass this divisibility test** because they are not even.

- 3
- -103
- 1.50
- 221

### Divisibility by 3 Rule

**Rule**: A number is divisible by 3 if the** sum of its digits** is divisible by 3.

See if the following number: is evenly divisible by three.

**Examples** of numbers that **are divisible** by 3

- 12→1 + 2 = 3 And 3 is divisible by 3 so the number 12 is also divisible by 3.
- 36→3 + 6 = 9 And 9 is divisible by 3 which means that 36 is also.
- 102→1 + 0 + 2 = 3
- 100,002,000 = 1 +0 +0 +0 +0 +2 +0 + 0 +0 = 3 so this very large number passes this divisibility test.
- 36 = 3 + 6 = 9 and we all know that 9 ÷ 3 = 1 so this number satisfies the rule and is evenly divided by 3!

**Examples**of numbers that do

**not pass this test**14→

**not divisible**by 3, so 14 is also not.

- 124→1 + 2+ 4 = 7 which is no good, does not work.
- 100,002,001 = 1 +0 +0 +0 +0 +2 +0 + 0 + 1= 4 so this very large also does not pass this divisibility test.

### Divisibility by 4 Rule

**Rule**: A number is divisible by 4 if the number's **last two digits** are divisible by 4.

Use the divisibility calculator below to determine if any number is divisible by four. Type in any number that you want, and the calculator will use the rule for divisibility by 4 to explain the result.

See if the following number: is evenly divisible by four

**Examples** of numbers that **are divisible** by 4

- 112 → since the last two digits, 12,
**are divisible**by 4, the number 112 is satisfies this rule and is also divisible by 4. - 10,948 → the last two digits, 48,
**are divisible**by 4. Therefore, the whole number is also. - 100,002,088 = 88. Yep, this satisfies rule because 88 is divisible by 4!
- -12,036 = 36 and 36 is evenly divided by 4, so -12,036 passes the test!

**Examples** of numbers that are **do not pass this divisibility test**

- 113 → since the last two digits, 13,
**are not divisible**by 4, the whole number does not pass this divisibility test. - 10,941 → the last two digits, 41,
**are not de visible**by 4. Therefore, the whole number does not satisfy the rule for 4. - 100,002,014 = 14 and 14 is no good, does not work.
- -1,011 = 11 so 1,011
**fails this test**.

Ever wonder why these rules work. The test for 4 makes sense if you just break down the numbers. Think about what this rule says: "All that matters is whether or not the last two digits are divisible by 4." Let's look at why this rule is true.

**Examine some three digit numbers**

- 124 is the same as 100 + 24, and we know that 100 is divisible by 4 so all that matters here is whether or not 24, or the last two digits, are divisible by 4. The same could be said for any three digit number 224 = 200 + 24, and we know that 200 is divisible by 4 so again all that we're worried about are these last two digits.

### Divisibility by 5 Rule

(another well known rule)**Rule**: A number is divisible by 5 if the its **last digit** is a 0 or 5.

Use the divisibility calculator below to determine if any number is divisible by five. Type in any number that you want, and the calculator will use the rule for divisibility by 5 to explain the result.

See what the rule for divisibility by five has to say about the following number:

**Examples** of numbers that **are divisible** by 5 and satisfy this rule

- 10 → since the last digit is 0, 10 satisfies this rule and is divisible by 5
- 15 → since the last digit is 5, 15 satisfies this rule and is divisible by 5
- 45
- -30
- 55
- -105
- 12,340

**Examples** of numbers that **fail this divisibility test**.

- 17 → since the last digit is 7, 17 does not satisfy this rule and is
**not**divisible by 5 - 118 → since the last digit is 8, 118 does not satisfy this rule and is
**not**divisible by 5 - -311 → Since the last digit is 1, 311 does not satisfy the rule for 5
- -101
- 12,103

### Divisibility by 6 Rule

Since 6 is a multiple of 2 and 3, the rules for divisibility by 6 are a combination of the rule for 2 and the rule for 3. In other words, a number passes this divisibility test only if it passes the test for 2 **and** test for 3.

Use the divisibility calculator below to determine if any number is divisible by six. Type in any number that you want, and the calculator will use the rule for divisibility by 6 to explain the result.

See if the following number: is evenly divisible by six

**Rule** A number is divisible by 6 if it **is even and** if the

**sum of its digits**is divisible by 3.

**Examples** of numbers that **are divisible** by 6

- 12 → satisfies both conditions:
- 114 → satisfies both conditions
- 1) 1+1+4 = 6 which is divisible by 3
- 2) 114 is even

- 241,122 → This passes the test because it's even and the sum of its digits can be evenly divided by 3.

- 207 → Fails the test since it's not even. We don't even have to see whether the second condition is satisfied since both conditions must be satisfied to pass this test. If only one of the two conditions (divisible by 2 and by 3) are not met, then the number does not satisfy the rule for 6.
- 241,124 → Although this number is even, the sum of its digits are not evenly divided by 6 so this fails the test.

**Examples**of numbers that are

**do not pass this divisibility test**

### Divisibility by 8 Rule

**Rule** A number passes the test for 8 if the** last three digits **form a number is divisible 8.

Use the divisibility calculator below to determine if any number is divisible by eight. Type in any number that you want, and the calculator will use the rule for divisibility by 8 to explain the result.

See what the rule for divisibility by eight has to say about the following number:

**Examples** of numbers that satisfy this rule and **are divisible** by 8

- 9,640 → 640 ÷ 8 = 80 so the whole number, 9,640, is divisible by 8
- 77, 184 → 184 ÷ 8 = 23 so 77,184 passes this divisibility test.
- 67, 536 → 536 is divisible by 8 ( 536 ÷ 8 = 67) so 67,536, is also.
- -30 → 640 ÷ 8 = 80 so the whole number, 9640, passes this test.
- 20,233,322,496 → Well, maybe you were wondering if this divisibility rule was really helpful or not. Once you get a giant number like 20,233,322,496, you start to realize what a nice trick this is to have up your sleeve! All you have to do is divide 496 by 8 to learn that the entire number is divisible by 8.
- - 316,145,664 → 664 passes this divisibility test.

**Examples** of numbers that are **do not pass this divisibility test**

- 9,801 → since 801 is not divisible by 8, 9,801 is not.
- 234,516 → Nope, no good. 516 is not evenly divided by 8 so the whole number fails the test!
- -32,344,588 → 588 does not work, so -32,344,588 does not satisfy the rule for 8!

### Divisibility by 9 Rule

**Rule** A number is divisible by 9 if the** sum of the digits **are evenly divisible 9.

Use the divisibility calculator below to determine if any number is divisible by nine. Type in any number that you want, and the calculator will explain whether or not it's divisible by nine based on this rule.

See if the following number: is evenly divisible by nine

**Examples** of numbers that satisfy this rule and **are divisible** by 9

- 4,518 → 4+5+1+8=18 and since 18 ÷9 = 2 , the whole number, 4,518, is divisible by 9
- 7,209 → 7+2+0+9 = 18, and by the same logic of the prior example, 7,209 passes this divisibility test.
- 6,993 → ,6993 is divisible by 9(6+9+9+3 = 27 & 27 ÷ 9 = 3) so 6,993 satisfies the rule for 9.
- 10,006,470 → Well, maybe you were wondering if this divisibility trick was really helpful or not. Once you get a giant number like 10,006,470, you start to realize what a nice trick this is to have up your sleeve! All you have to do is add the digits (1+ 6+4+7 = 18) to quickly see that the entire number is divisible by 9 (18÷9 = 2).

**Examples** of numbers that are **do not pass this divisibility test**

- 29 → 2+9 =11. Since 11 is not divisible by 9, 29 is not either.
- 6,992 → Nope, no good. 6+9+9+2 =26 which is not evenly divided by 9 so the whole number fails the test!

### Divisibility by 10 Rule

**Rule** A number passes the test for 10 if its final digit is 0

Use the divisibility calculator below to determine if any number is divisible by ten. Type in any number that you want, and the calculator will use the rule for divisibility by 10 to explain the result.

See what the rule for divisibility by ten has to say about the following number:

**Examples** of numbers that are **do not pass this divisibility test**

- 91,801 → last digit is not zero, so this does not work.
- 234,516 → Nope, this number does not satisfy the rule for 10.
- -32,344,508 → Again, it all comes down to that last digit which just has to be zero!

### Divisibility by 11 Rule

**Rule** A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.(This abstract and confusing sounding rule is much clearer with a few examples)

Use the divisibility calculator below to determine if any number is divisible by eleven. Type in any number that you want, and the calculator will explain whether or not it's divisible by 11 based on this rule.

See if the following number: is evenly divisible by eleven

**Examples** of numbers that satisfy this rule

- 946 → (9+6) - 4 = 11 which is, of course, evenly divided by 11 so 946 passes this divisibility test
- 10,813 → (1+8+3) - (0+1) = 12-1 =11. Yes, this satisfies the rule for 11!
- 25, 784 = → (2+ 7 + 4) - (5+8) = 13 - 13 =0 . Yes, this does indeed work. In case you found this last bit confusing, remember that any number evenly divides 0. Think about it, how many 11's are there in 0? None, right. Well that means that 11 divides zero, zero times!
- 119,777,658 → (1+ 9 + 7 + 6 + 8) - (1+ 7 + 7 +5) = 31 - 20 = 11

**do not pass this divisibility test**

- 947 → (9+7) - 4 = 12 which is not divisible by 11
- 10,823 → (1+8+3) - (0+2) = 12- 2 =10. No, no good. This one fails!
- 35, 784 = → (3 + 7 + 4) - (5+8) = 14 - 13 = 1. No, does not satisfy the rule for 11!
- 12,347, 496, 132 = → (1+3+7+9+3) - (2 + 4 +4 + 6 + 3)= 23- 19 = 4

Well, at the top of this page, I told you that some rules are not actually painful at all. Some of the rules for divisibility on this page actually work better than the best graphing calculators that you can buy!