Use divisibility rules for 3 and 9

In base-ten whole numbers, divisibility by 3 or 9 can be determined by adding the digits: a number is divisible by 3 exactly when its digit sum is divisible by 3, and divisible by 9 exactly when its digit sum is divisible by 9. This supports identifying factors and multiples and distinguishing that every multiple of 9 is a multiple of 3, but not every multiple of 3 is a multiple of 9; formal modular arithmetic and generalized proofs are beyond this scope.

Detailed Explanation: Use divisibility rules for 3 and 9

To test whether a whole number is divisible by $3$ or $9$ , add its digits.

If the digit sum is divisible by $3$ , the number is divisible by $3$ .
If the digit sum is divisible by $9$ , the number is divisible by $9$ .

Example: Is $(2{,}145)$ divisible by $3$ or $9$ ?

Step 1: Add the digits.

2+1+4+5=12

Step 2: Check $(12)$ .

$(12)$ is divisible by $3$ , because $(12=3\times4)$ .
$(12)$ is not divisible by $9$ .

Step 3: State the answer.

Therefore, $(2{,}145)$ is divisible by $3$ , but it is not divisible by $9$ .

Remember: Every number divisible by $9$ is also divisible by $3$ , but a number divisible by $3$ is not always divisible by $9$ .

Learn by doing: Use divisibility rules for 3 and 9

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Divisibility Rules (Easy) - Dividend and Condition to Yes/No

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