Divisibility rules for 3, 6, and 9

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...

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...

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...