Lesson 21: Divisibility

New Concept

A number is divisible by another if it can be divided without a remainder. These rules are shortcuts to find factors without division.

Last-Digit Tests
A number is divisible by:

• $2$ if the last digit is even.
• $5$ if the last digit is $0$ or $5$.
• $10$ if the last digit is $0$.

Sum-of-Digits Tests
A number is divisible by:

• $3$ if the sum of the digits is divisible by $3$.
• $9$ if the sum of the digits is divisible by $9$.

What’s next

Next, you will apply these rules in worked examples and practice problems to quickly test any number for its factors.

Property

Inspect the last digit of the number. A number is divisible by:

• $2$ if the last digit is even.
• $5$ if the last digit is $0$ or $5$.
• $10$ if the last digit is $0$.

Examples

• $2468$ is divisible by $2$ because its last digit, $8$, is an even number.
• $975$ is divisible by $5$ because its last digit is $5$.
• $3450$ is divisible by both $5$ and $10$ because its last digit is $0$.

Explanation

Forget long division! To check for divisibility by $2$, $5$, or $10$, just peek at the very last digit. It's the ultimate shortcut. If the last digit gives you the right signal—like being even for the number $2$—you have cracked the code without doing any heavy math lifting. It's the sneakiest and fastest test in the book!

Property

Add the digits of the number and inspect the total. A number is divisible by:

• $3$ if the sum of the digits is divisible by $3$.
• $9$ if the sum of the digits is divisible by $9$.

Examples

• Is $762$ divisible by $3$? Yes, because $7+6+2=15$, and $15$ is divisible by $3$.
• Is $2745$ divisible by $9$? Yes, because $2+7+4+5=18$, and $18$ is divisible by $9$.
• Is $128$ divisible by $3$? No, because $1+2+8=11$, and $11$ is not divisible by $3$.

Explanation

This test is like a magic trick! To see if a big number is divisible by $3$ or $9$, just add up all its individual digits. If that little sum is divisible by $3$ or $9$, then the original huge number is too! It is a simple way to handle large numbers without breaking a sweat. Teamwork makes the dream work!