## 1. Divisibility by 2

This is the easiest of divisibility rules and the very first one to use when checking divisibility of a number or if a number is prime.

A number is divisible by 2 if it ends in an even number. So, if the number ends in 0, 2, 4, 6, or 8, it’s divisible by 2.

**More Examples:**

- 10222 – Yes, because it ends with a 2.
- 8,738,789 – Nope, ends with a 9.
- 783,897 – Again, no, ends with a 7.
- 93757374 – Yes! Ends with a 4.

Easy, right?

## 2. Divisibility by 3

This one’s surprisingly easy too!

Add up all the digits of the number. Is the sum divisible by 3? Then the entire number is divisible by 3!

**More Examples:**

**98,71,654,323:**- Add up all digits: 9 + 8 + 7 + 1 + 6 + 5 + 4 + 3 + 2 + 3 = 48.
- Add digits again: 4 + 8 = 12.
- 12/3 = 4
- Yes, 98,71,654,323 is divisible by 3.

**85,490**- Add up all digits: 8 + 5 + 4 + 9 + 0 = 26
- Add digits again: 2+6=8
- 8 is not divisible by 3.
- No, 85,490 is not divisible by 3.

## 3. Divisibility by 4

Remember divisibility by 2? This one is very similar! To be divisible by 4, the last 2 digits (units and tens place) of the number should be divisible by 4.

**More Examples:**

**98,71,654,324:**- Last digit even? Yes.
- Last 2 digits divisible by 4? 24/4 = 6
- Yes, 98,71,654,324 is divisible by 4.
- Easy, huh?

**85,492**- Last digit even? Yes.
- Last 2 digits divisible by 4? 92/4 = 23
- Yes, 85,492 is not divisible by 4.

## 4. Divisibility by 5

This one’s super-easy too, no pesky additions or divisions. Is the last (units) digit a 5 or a 0? Wham, it’s divisible by 5!

**More Examples:**

**98,71,654,320:**- Last (units) digit 5 or 0? Yes.
- Yes, 98,71,654,320 is divisible by 5.

**85,492**- Last (units) digit 5 or 0? No.
- No, 85,492 is not divisible by 5.

## 5. Divisibility by 6

While not as easy as some of the others, this one combines 2 easy rules – divisibility by 2 and 3. This makes sense if you think about the fact that 2 x 3 = 6.

- Check 1 – is it divisible by 2 ⇒ Are last (units) digit even (0, 2, 4, 6, 8)?
- If yes, Check 2 – is it divisible by 3 ⇒ Is sum of all digits divisible by 3?

**More Examples:**

**98,71,654,3023:**- Last (units digit) even? No.
- No, 98,71,654,323 is not divisible by 6.

**85,494**- Is last (units) digit even? Yes
- Add up all digits: 8 + 5 + 4 + 9 + 4 = 30
- 30/3 = 10.
- Yes, 85,494 is divisible by 3.

## 6. Divisibility by 7

This one’s actually fun with some practice, I promise! And makes you feel very smart for knowing such a cool trick!

- Multiply last (units) digit by 2.
- Now subtract from rest of the number.
- Rinse, repeat until number is small enough
- Is final number 0 or multiple of 7? Then the original number is divisible by 7

**More Examples:**

**9,876,538 (this one is long and tedious and it may be easier to just divide by 7 but I wanted to show you a full method):**- Units digit x 2 (8 x 2 = 16)
- 987653 – 16 = 987637
- 7 x 2 = 14
- 98763 – 14 = 98749
- 9 x 2 = 18
- 9874 – 18 = 9856
- 6 x 2 = 12
- 985 – 12 = 973
- 3 x 2 = 6
- 97 – 6 = 91
- 1 x 2 = 2
- 9 -2 = 7
- Is this simplified number divisible by 7? Yes!
- Yes, 9,876,538 is divisible by 7.

**31,696**- 6 x 2 = 12
- 3169 – 12 = 3157
- 7 x 2 = 14
- 315 – 14 = 301
- 1 x 2 = 2
- 30-2 = 28
- 28/7 = 4.
- Yes, 31,696 is divisible by 7.

## 7. Divisibility by 8

Remember the divisibility rule for 4? This one is almost the same. If the last three digits of a whole number are divisible by 8, then the entire number is divisible by 8.

More Examples:

**9,876,538:**- 538/8 = 67 with a remainder of 2.
- No, 9,876,538 is not divisible by 8.

**31,696**- 696/8 = 87.
- Yes, 31,696 is divisible by 8.

## 8. Divisibility by 9

Remember the divisibility rule for 3? The same, with a small twist! Add up all the digits of the number. Is the sum divisible by 9? Then the entire number is divisible by 9!

More Examples:

**98,71,654,320:**- Add up all digits: 9 + 8 + 7 + 1 + 6 + 5 + 4 + 3 + 2 + 0 = 45.
- 45/9 = 5.
- Yes, 98,71,654,320 is divisible by 9.

**85,490**- Add up all digits: 8 + 5 + 4 + 9 + 0 = 26
- 26 is not divisible by 9.
- No, 85,490 is not divisible by 9.

## 9. Divisibility by 10

This one is almost something that we learn in elementary school and doesn’t need to be explained. Does the number end with a 0? If yes, it’s divisible by 10. That’s it!