Divisibility Rules (2–10)
Definitions, Worked Examples, and Interactive Practice
What is Divisibility?
A whole number \( n \) is said to be divisible by another whole number \( m \) if the division: \[ \frac{n}{m} \] results in a remainder of \(0\). In this case, \( m \) is called a factor of \( n \).
Example:
\[ \frac{15}{5} = 3 \]
Since the remainder is \(0\), the number \(15\) is divisible by \(5\). This can also be written as:
\[ 15 = 3 \times 5 \]
Both \(3\) and \(5\) are factors of \(15\). Similarly, since \( \frac{15}{3} = 5 \), \(15\) is also divisible by \(3\).
Why use divisibility rules? These shortcuts let you check whether a number is divisible by another without performing long division. They are especially useful for mental math, simplifying fractions, and factoring.
Divisibility Tests (Rules 2 through 10)
Given a whole number, the following rules help quickly determine whether it is divisible by another number without doing long division.
Divisible by 2
Rule: A whole number is divisible by \(2\) if its last digit is even (\(0, 2, 4, 6, \text{ or } 8\)).
Example: The number \(23,464,56\mathbf{8}\) is divisible by \(2\) because its last digit is \(8\).
Why it works: Since \(10\) is divisible by \(2\), every place value beyond the units digit is also divisible by \(2\). Only the last digit matters.
Divisible by 3
Rule: A whole number is divisible by \(3\) if the sum of its digits is divisible by \(3\).
Example: For the number \(1272\): \[ 1 + 2 + 7 + 2 = 12 \] Since \(12\) is divisible by \(3\), the number \(1272\) is divisible by \(3\).
Why it works: Since \(10 \equiv 1 \pmod 3\), any power of \(10\) is also congruent to \(1 \pmod 3\). Therefore, a number is congruent to the sum of its digits modulo \(3\).
Divisible by 4
Rule: A whole number is divisible by \(4\) if the number formed by its last two digits is divisible by \(4\).
Example: The last two digits of \(1,869,5\mathbf{20}\) form the number \(20\). Since \( \frac{20}{4} = 5 \), the number \(1,869,520\) is divisible by \(4\).
Why it works: Since \(100\) is divisible by \(4\), every place value from the hundreds digit upward is divisible by \(4\). Only the last two digits matter.
Divisible by 5
Rule: A whole number is divisible by \(5\) if its last digit is \(0\) or \(5\).
Example: The number \(52,74\mathbf{5}\) is divisible by \(5\) because its last digit is \(5\).
Why it works: Since \(10\) is divisible by \(5\), every place value beyond the units digit is divisible by \(5\). Only the last digit matters.
Divisible by 6
Rule: A whole number is divisible by \(6\) if it is divisible by both \(2\) and \(3\).
Example: The number \(1890\) is divisible by \(2\) (it ends in \(0\)). It is also divisible by \(3\) because \(1 + 8 + 9 + 0 = 18\), and \(18\) is a multiple of \(3\). Therefore, \(1890\) is divisible by \(6\).
Why it works: Since \(6 = 2 \times 3\) and \(2\) and \(3\) are coprime, a number is divisible by \(6\) if and only if it is divisible by both \(2\) and \(3\).
Divisible by 7 (Quick Rule)
Rule: Double the last digit and subtract it from the remaining number. If the result is \(0\) or a multiple of \(7\), the original number is divisible by \(7\).
Example: For \(154\): \(15 - 2(4) = 15 - 8 = 7\), which is a multiple of \(7\). So \(154\) is divisible by \(7\).
Divisible by 8
Rule: A whole number is divisible by \(8\) if the number formed by its last three digits is divisible by \(8\).
Example: The last three digits of \(18,567,\mathbf{160}\) form \(160\). Since \( \frac{160}{8} = 20 \), the entire number \(18,567,160\) is divisible by \(8\).
Why it works: Since \(1000\) is divisible by \(8\), every place value from the thousands digit upward is divisible by \(8\). Only the last three digits matter.
Divisible by 9
Rule: A whole number is divisible by \(9\) if the sum of its digits is divisible by \(9\).
Example: For the number \(1006506\): \[ 1 + 0 + 0 + 6 + 5 + 0 + 6 = 18 \] Since \(18\) is divisible by \(9\), the number \(1,006,506\) is divisible by \(9\).
Why it works: Since \(10 \equiv 1 \pmod 9\), any power of \(10\) is also congruent to \(1 \pmod 9\). Therefore, a number is congruent to the sum of its digits modulo \(9\).
Divisible by 10
Rule: A whole number is divisible by \(10\) if its last digit is \(0\).
Example: The number \(12,635,36\mathbf{0}\) is divisible by \(10\) because it ends in \(0\).
Why it works: Since \(10\) is the base of our number system, any number ending in \(0\) is a multiple of \(10\).
Interactive Practice Questions
Test your knowledge! Select the numbers from each list that satisfy the rule. Click "View Solution" to check your answers.
Part A: Which numbers are divisible by 2?
List: \(4,\; 5,\; 100,\; 408,\; 777,\; 52340,\; 8879\)
Solution: \(4,\; 100,\; 408,\; 52340\) (They all end in \(0, 2, 4, 6,\) or \(8\))
Part B: Which numbers are divisible by 3?
List: \(33,\; 53,\; 105,\; 5554,\; 777,\; 9222321\)
Solution: \(33,\; 105,\; 777,\; 9222321\) (The sum of their digits is a multiple of \(3\))
Part C: Which numbers are divisible by 4?
List: \(100,\; 3005,\; 12940,\; 5554,\; 7777,\; 9222352\)
Solution: \(100,\; 12940,\; 9222352\) (Their last two digits—\(00, 40,\) and \(52\)—are divisible by \(4\))
Part D: Which numbers are divisible by 5?
List: \(60,\; 362,\; 12940,\; 50016505\)
Solution: \(60,\; 12940,\; 50016505\) (They end in \(0\) or \(5\))
Part E: Which numbers are divisible by 6?
List: \(120,\; 648,\; 12941,\; 309948\)
Solution: \(120,\; 648,\; 309948\) (They are even numbers whose digits sum to a multiple of \(3\))
Part F: Which numbers are divisible by 8?
List: \(160,\; 11048,\; 12941,\; 10056720\)
Solution: \(160,\; 11048,\; 10056720\) (Their last three digits form a number divisible by \(8\))
Part G: Which numbers are divisible by 9?
List: \(109,\; 10233,\; 12941,\; 1946700\)
Solution: \(10233,\; 1946700\) (The sum of their digits is a multiple of \(9\))
Part H: Which numbers are divisible by 10?
List: \(100,\; 3005,\; 12940\)
Solution: \(100,\; 12940\) (They end in \(0\))
Frequently Asked Questions
What is the divisibility rule for 7?
The rule for 7 is: double the last digit and subtract it from the remaining number. If the result is 0 or a multiple of 7, the original number is divisible by 7. For a full guide, see our dedicated divisibility rule for 7 page.
Why do the divisibility rules for 3 and 9 work?
They work because \(10 \equiv 1 \pmod 3\) and \(10 \equiv 1 \pmod 9\). This means any power of 10 leaves a remainder of 1 when divided by 3 or 9. Therefore, a number is congruent to the sum of its digits modulo 3 (or 9). If that sum is divisible by 3 (or 9), the original number is too.
Can a number be divisible by 6 but not by 3?
No. Since 6 = 2 × 3, any number divisible by 6 must also be divisible by both 2 and 3. Divisibility by 6 requires divisibility by 3.
What is the difference between a factor and a divisor?
They are essentially the same thing. If \(n\) is divisible by \(m\), then \(m\) is both a factor and a divisor of \(n\).