Use divisibility rules for 2, 3, 5, 9, and 10

A whole number is divisible by 2, 5, or 10 based on its ones digit: it must be even, end in 0 or 5, or end in 0, respectively. Divisibility by 3 or 9 is determined by whether the sum of the decimal digits is divisible by 3 or 9, connecting these tests to place value and the structure of multiples. The scope is limited to these rules for decimal whole numbers, not generalized divisibility tests or modular arithmetic.

Detailed Explanation: Use divisibility rules for 2, 3, 5, 9, and 10

A number is divisible by another number if it can be divided evenly, with no remainder.

Use these rules:

Divisible by 2: The ones digit is even: $0,2,4,6,$ or $8$ .
Divisible by 3: The sum of the digits is divisible by $3$ .
Divisible by 5: The ones digit is $0$ or $5$ .
Divisible by 9: The sum of the digits is divisible by $9$ .
Divisible by 10: The ones digit is $0$ .

Worked example

Which of $2,3,5,9,$ and $10$ divide $4{,}350$ evenly?

Step 1: Test divisibility by $2$ .

The ones digit of $4{,}350$ is $0$ . Since $0$ is even, $4{,}350$ is divisible by $2$ .

Step 2: Test divisibility by $3$ and $9$ .

Add the digits:

4+3+5+0=12

Since $12$ is divisible by $3$ , $4{,}350$ is divisible by $3$ .

Since $12$ is not divisible by $9$ , $4{,}350$ is not divisible by $9$ .

Step 3: Test divisibility by $5$ .

The ones digit is $0$ , so $4{,}350$ is divisible by $5$ .

Step 4: Test divisibility by $10$ .

The ones digit is $0$ , so $4{,}350$ is divisible by $10$ .

Therefore, $4{,}350$ is divisible by

2,\ 3,\ 5,\text{ and }10,

but it is not divisible by $9$ .

Learn by doing: Use divisibility rules for 2, 3, 5, 9, and 10

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Divisibility Rules (Medium) - Dividend to Yes/No

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