Mathematics
Grade 11
15 min
Divisibility rules for 3, 6, and 9
Divisibility rules for 3, 6, and 9
What you'll learn
- Identify numbers less than 100 that are divisible by 3 by determining if the sum of their digits is divisible by 3 with 80% accuracy.
- Explain in their own words why a number is divisible by 9 if the sum of its digits is divisible by 9, providing at least one example.
- Solve at least 3 out of 4 problems correctly, determining if a given number less than 100 is divisible by both 3 and 2, and therefore divisible by 6.
- Apply the divisibility rules for 3, 6, and 9 to sort a set of 10 numbers (less than 100) into the correct categories with at least 70% accuracy.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the divisibility rules for 3, 6, and 9 and apply them to integers and variable expressions.
Represent a multi-digit number as a polynomial expression with a base of 10.
Algebraically prove the divisibility rules for 3 and 9 using the polynomial representation of integers.
Solve for an unknown digit within a variable expression to satisfy given divisibility conditions for 3, 6, or 9.
Analyze and solve problems that require the application of multiple divisibility rules simultaneously.
Differentiate between the necessary conditions for divisibility by 3, 6, and 9, avoiding common misconceptions.
How can you determine if 8,547,392,475,912 is divisible by 3 in under 10 seconds without a calculator? 🤯 Let's find out!
This tutorial moves beyond sim...
2
Key Concepts & Vocabulary
TermDefinitionExample
DivisibilityAn integer 'a' is divisible by an integer 'b' if dividing 'a' by 'b' results in a remainder of 0. We can write this as b | a.24 is divisible by 3 because 24 ÷ 3 = 8 with a remainder of 0. So, 3 | 24.
Sum of DigitsThe result of adding together all the individual digits of a number.For the number 582, the sum of its digits is 5 + 8 + 2 = 15.
Variable Expression (in this context)A representation of an integer where one or more digits are replaced by variables. The variable represents a single digit from 0 to 9.The expression '7k4' represents the three-digit integer 700 + 10k + 4, where 'k' is a digit.
Polynomial Representation of an IntegerExpressing an integer as a sum of its digits multiplied by p...
3
Core Formulas
Divisibility Rule for 3 and 9
Let N be an integer and S be the sum of its digits. N is divisible by 3 if and only if S is divisible by 3. N is divisible by 9 if and only if S is divisible by 9. Algebraically: N \equiv S \pmod{3} and N \equiv S \pmod{9}
This is the most fundamental rule. To test a number for divisibility by 3 or 9, you don't need to divide the number itself. Instead, sum its digits and test the sum, which is a much smaller, more manageable number.
Divisibility Rule for 6
An integer N is divisible by 6 if and only if it is divisible by BOTH 2 and 3. (N \equiv 0 \pmod{2}) \land (N \equiv 0 \pmod{3})
This is a compound rule. It requires checking two separate conditions. First, the number must be even (its last digit is 0, 2, 4, 6, or 8). Second, the sum of...
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
What is the largest 4-digit number of the form '8X6Y' that is divisible by 6?
A.8968
B.8964
C.8868
D.8766
Challenging
A 6-digit number '14A5B2' is divisible by 9. How many different ordered pairs of digits (A, B) are possible?
A.11
B.9
C.10
D.12
Challenging
The 5-digit number N = 4x5y6 is divisible by both 6 and 9 (i.e., divisible by 18). What is the value of x + y?
A.3
B.12
C.3 or 12
D.6 or 15
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Variable expressions
Mathematics for other grades
Frequently asked questions
What grade level is "Divisibility rules for 3, 6, and 9"?
Divisibility rules for 3, 6, and 9 is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Divisibility rules for 3, 6, and 9?
You'll be able to: Identify numbers less than 100 that are divisible by 3 by determining if the sum of their digits is divisible by 3 with 80% accuracy; Explain in their own words why a number is divisible by 9 if the sum of its digits is….
Is "Divisibility rules for 3, 6, and 9" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Divisibility rules for 3, 6, and 9?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.