# Divisibility Rules: 2, 3, 4, 5, 6, 9, and 10

A number [latex]a[/latex] is divisible by the number [latex]b[/latex] if [latex]a \div b[/latex] has a remainder of zero ([latex]0[/latex]). For example, 15 divided by 3 is exactly 5 which implies that its remainder is zero. We then say that 15 is divisible by 3.

In our other lesson, we discussed the divisibility rules for 7, 11, and 12. This time, we will cover the divisibility rules or tests for** 2**, **3**, **4**, **5**, **6**, **9**, and **10**. Believe me, you will be able to learn them very quickly because you may not know that you already have a basic and intuitive understanding of it. For instance, it is obvious that all even numbers are divisible by 2. That is pretty much the divisibility rule for **2**. The goal of this divisibility rules lesson is to formalize what you already know.

Divisibility rules help us to determine if a number is divisible by another without going through the actual division process such as the long division method. If the numbers in question are numerically small enough, we may not need to use the rules to test for divisibility. However, for numbers whose values are large enough, we want to have some rules to serve as “shortcuts” to help us figure out if they are indeed divisible by each other.

## Divisibility Rules for Numbers 2, 3, 4, 5, 6, 9 and 10

**A number is divisible by 2 if the last digit of the number is 0, 2, 4, 6, or 8.**

Example 1: Is the number 246 divisible by 2?

Solution: Since the last digit of the number 246 ends in 6, that means it is divisible by 2.

Example 2: Which of the numbers 100, 514, 309, and 768 are divisible by 2?

Solution: If we examine all four numbers, only the number 309 doesn’t end with 0, 2, 4, 6, or 8. We can conclude that all the numbers above except 309 are divisible by 2.

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

Example 1: Is the number 111 divisible by 3?

Solution:** **Let’s add the digits of the number 111. We have 1 + 1 + 1 = 3. Since the sum of the digits is divisible by the 3, therefore the number 111 is also divisible by 3.

Example 2: Which one of the two numbers 522 and 713 is divisible by 3?

Solution: The sum of the digits of 522 (5+2+2=9) is 9 which is divisible by 3. That makes 522 divisible by 3. However, the number 713 has 11 as the sum of its digits which is clearly not divisible by 3 thus 713 is not divisible by 3. Therefore, only 522 is divisible by 3.

**A number is divisible by 4 if the last two digits of the number are divisible by 4.**

Example 1: What is the only number in the set below is divisible by 4?

{945, 736, 118, 429}

Solution:** **Observe the last two digits of the four numbers in the set. Notice that 736 is the only number wherein the last two digits (36) is divisible by 4. We can conclude that 736 is the only number in the set that is divisible by 4.

Example 2: True or False. The number 5,554 is divisible by 4.

Solution:** **The last two digits of the number 5,554 is 54 which is not divisible by 4. That means the given number is NOT divisible by 4 so the answer is **false**.

**A number is divisible by 5 if the last digit of the number is 0 or 5.**

Example 1: Multiple Choice. Which number is divisible by 5.

*A)* 68

*B)* 71

*C)* 20

*D)* 44

Solution: In order for a number to be divisible by 5, the last digit of the number must be either 0 or 5. Going over the choices, only the number 20 is divisible by 5 so the answer is choice** C**.

Example 2: Select all the numbers that are divisible by 5.

*A)* 27

*B)* 105

*C)* 556

*D)* 343

E) 600

Solution: Both 105 and 600 are divisible by 5 because they either end in 0 or 5. Thus, options **B** and **E** are the correct answers.

**A number is divisible by 6 if the number is divisible by both 2 and 3.**

Example 1: Is the number 255 divisible by 6?

Solution: For the number 255 to be divisible by 6, it must divisible by 2 and 3. Let’s check first if it is divisible by 2. Note that 255 is not an even number (any number ending in 0, 2, 4, 6, or 8) which makes it not divisible 2. There’s no need to check further. We can now conclude that this is not divisible by 6. The answer is **NO**.

Example 2: Is the number 4,608 divisible by 6?

Solution: A number is an even number so it is divisible by 2. Now check if it is divisible by 3. Let’s do that by adding all the digits of 4,608 which is 4 + 6+ 0 + 8 = 18. Obviously, the sum of the digits is divisible by 3 because 18 ÷ 3 = 6. Since the number 4,608 is both divisible by 2 and 3 then it must also be divisible by 6. The answer is **YES**.

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

Example 1: Is the number 1,764 divisible by 9?

Solution: For a number to be divisible by 9, the sum of its digits must also be divisible by 9. For the number 1,764 we get 1 + 7 + 6 + 4 = 18. Since the sum of the digits is 18 and is divisible by 9 therefore 1,764 must be divisible by 9.

Example 2: Select all the numbers that are divisible by 9.

*A)* 7,065

*B)* 3,512

*C)* 8,874

*D)* 22,778

*E)* 48,069

Solution:** **Let’s add the digits of each number and check if its sum is divisible by 9.

- For 7,065, 7 + 0 + 6 + 5 = 18 which is divisible by 9.
- For 3,512, 3 + 5 + 1 + 2 = 11 which is
**NOT**divisible by 9. - For 8,874, 8 + 8 + 7 + 4 = 27 which is divisible by 9.
- For 22,778, 2 + 2 + 7 + 7 + 8 = 26 which is
**NOT**divisible by 9. - For 48,069, 4 + 8 + 0 + 6 + 9 = 27 which is divisible by 9.

Therefore, choices **A**, **C**, and **E** are divisible by 9.

**A number is divisible by 10 if the last digit of the number is 0.**

The numbers 20, 40, 50, 170, and 990 are all divisible by 10 because their last digit is zero, [latex]0[/latex]. On the other hand, 21, 34, 127, and 468 are not divisible by 10 since they don’t end with zero.

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