Extended Divisibility Rules

Divisibility rules are quick shortcuts that help you find out if one number can be divided evenly by another number without having to do long division. In earlier grades, you learned the basic rules. Now, let's extend our toolkit to include the rules for 4, 6, 8, and 9.

Quick Review of Basic Rules

Before we dive into the new rules, let's quickly review the basics:

• Divisible by 2: The number is even (ends in 0, 2, 4, 6, or 8).
• Divisible by 3: The sum of all the digits is divisible by 3.
• Divisible by 5: The number ends in 0 or 5.
• Divisible by 10: The number ends in 0.

The Extended Rules: 4, 6, 8, and 9

Divisibility by 4

The Rule: A number is divisible by 4 if its last two digits form a number that is divisible by 4.

Example: Which of $234$, $345$, and $456$ are divisible by 4?

• For $234$, the last two digits are $34$. Since $34$ is not divisible by 4, $234$ is not divisible by 4.
• For $345$, the last two digits are $45$. Since $45$ is an odd number, it cannot be divisible by 4.
• For $456$, the last two digits are $56$. Since $56 \div 4 = 14$, $456$ is divisible by 4.

Divisibility by 6

The Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

Example: Is $4{,}356$ divisible by 6?

• Check 2: The number ends in 6, which is even. So, yes for 2.
• Check 3: Add the digits: $4 + 3 + 5 + 6 = 18$. Since $18$ is divisible by 3, yes for 3.
• Because it passes both tests, $4{,}356$ is divisible by 6.

Divisibility by 8

The Rule: A number is divisible by 8 if its last three digits form a number that is divisible by 8.

Example: Check $7{,}120$.

• The last three digits are $120$.
• Since $120 \div 8 = 15$, the whole number $7{,}120$ is divisible by 8.

Divisibility by 9

The Rule: A number is divisible by 9 if the sum of its digits is divisible by 9. (This is very similar to the rule for 3!)

Example: Test if $7{,}290$ is divisible by 9.

• Add the digits: $7 + 2 + 9 + 0 = 18$.
• Since $18 \div 9 = 2$, $7{,}290$ is divisible by 9.

Summary Table

Keep this quick reference guide handy when testing numbers:

Divisor	Rule
4	The last 2 digits are divisible by 4.
6	The number is divisible by both 2 and 3.
8	The last 3 digits are divisible by 8.
9	The sum of the digits is divisible by 9.