What are the Divisibility Rules

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A number is divisible by another number when division leaves no remainder. For example, \(24\) is divisible by \(6\) because \(24 \div 6 = 4\), but \(24\) is not divisible by \(5\) because there is a remainder.

Divisibility rules help you test a number quickly, often without long division.

Common Divisibility Rules

Divisible by \(2\): the last digit is even: \(0,2,4,6,\) or \(8\).
Divisible by \(3\): the sum of the digits is divisible by \(3\).
Divisible by \(4\): the number formed by the last two digits is divisible by \(4\).
Divisible by \(5\): the last digit is \(0\) or \(5\).
Divisible by \(6\): the number is divisible by both \(2\) and \(3\).
Divisible by \(7\): double the last digit and subtract it from the remaining number; if the result is divisible by \(7\), the original number is divisible by \(7\). Repeat if needed.
Divisible by \(8\): the number formed by the last three digits is divisible by \(8\).
Divisible by \(9\): the sum of the digits is divisible by \(9\).
Divisible by \(10\): the last digit is \(0\).
Divisible by \(11\): the alternating sum of the digits is divisible by \(11\), including \(0\).

Example: To test \(49\) for divisibility by \(7\), double the last digit: \(9 \times 2 = 18\). Subtract from the remaining number: \(4 - 18 = -14\). Since \(-14\) is divisible by \(7\), \(49\) is divisible by \(7\).

Divisibility Rules

Think of this lesson as more than a rule to memorize. Divisibility Rules is about number sense, equivalent forms, and careful arithmetic. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Fractions compare a part to a whole. Keep track of the numerator, denominator, and whether the pieces are the same size before adding, subtracting, or simplifying.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Divisibility Rules

1) Is \(246\) divisible by \(2\)?

2) Is \(319\) divisible by \(2\)?

3) Is \(852\) divisible by \(3\)?

4) Is \(1001\) divisible by \(3\)?

5) Is \(1316\) divisible by \(4\)?

6) Is \(5722\) divisible by \(4\)?

7) Is \(9825\) divisible by \(5\)?

8) Is \(7642\) divisible by \(5\)?

9) Is \(438\) divisible by \(6\)?

10) Is \(524\) divisible by \(6\)?

11) Is \(203\) divisible by \(7\)?

12) Is \(348\) divisible by \(7\)?

13) Is \(5816\) divisible by \(8\)?

14) Is \(7314\) divisible by \(8\)?

15) Is \(7290\) divisible by \(9\)?

16) Is \(3241\) divisible by \(9\)?

17) Is \(45870\) divisible by \(10\)?

18) Is \(45875\) divisible by \(10\)?

19) Is \(918082\) divisible by \(11\)?

20) Is \(123456\) divisible by \(11\)?

Answers

1) Is \(246\) divisible by \(2\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(2\).
Step 2: The last digit is \(6\), which is even.
Step 3: Therefore, \(246\) is divisible by \(2\).

2) Is \(319\) divisible by \(2\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(2\).
Step 2: The last digit is \(9\), which is odd.
Step 3: Therefore, \(319\) is not divisible by \(2\).

3) Is \(852\) divisible by \(3\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(3\).
Step 2: Add the digits: \(8+5+2=15\). Since \(15\) is divisible by \(3\), the number passes the rule.
Step 3: Therefore, \(852\) is divisible by \(3\).

4) Is \(1001\) divisible by \(3\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(3\).
Step 2: Add the digits: \(1+0+0+1=2\). Since \(2\) is not divisible by \(3\), the number fails the rule.
Step 3: Therefore, \(1001\) is not divisible by \(3\).

5) Is \(1316\) divisible by \(4\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(4\).
Step 2: The last two digits are \(16\), and \(16\) is divisible by \(4\).
Step 3: Therefore, \(1316\) is divisible by \(4\).

6) Is \(5722\) divisible by \(4\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(4\).
Step 2: The last two digits are \(22\), and \(22\) is not divisible by \(4\).
Step 3: Therefore, \(5722\) is not divisible by \(4\).

7) Is \(9825\) divisible by \(5\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(5\).
Step 2: The last digit is \(5\).
Step 3: Therefore, \(9825\) is divisible by \(5\).

8) Is \(7642\) divisible by \(5\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(5\).
Step 2: The last digit is \(2\), not \(0\) or \(5\).
Step 3: Therefore, \(7642\) is not divisible by \(5\).

9) Is \(438\) divisible by \(6\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(6\).
Step 2: The number is even, so it is divisible by \(2\). Its digit sum is \(4+3+8=15\), which is divisible by \(3\). It passes both tests.
Step 3: Therefore, \(438\) is divisible by \(6\).

10) Is \(524\) divisible by \(6\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(6\).
Step 2: The number is even, but the digit sum is \(5+2+4=11\), which is not divisible by \(3\). It must pass both the \(2\) and \(3\) tests.
Step 3: Therefore, \(524\) is not divisible by \(6\).

11) Is \(203\) divisible by \(7\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(7\).
Step 2: Double the last digit: \(3 \times 2=6\). Subtract from the remaining number: \(20-6=14\). Since \(14\) is divisible by \(7\), the number passes the rule.
Step 3: Therefore, \(203\) is divisible by \(7\).

12) Is \(348\) divisible by \(7\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(7\).
Step 2: Double the last digit: \(8 \times 2=16\). Subtract from the remaining number: \(34-16=18\). Since \(18\) is not divisible by \(7\), the number fails the rule.
Step 3: Therefore, \(348\) is not divisible by \(7\).

13) Is \(5816\) divisible by \(8\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(8\).
Step 2: The last three digits are \(816\), and \(816 \div 8 = 102\).
Step 3: Therefore, \(5816\) is divisible by \(8\).

14) Is \(7314\) divisible by \(8\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(8\).
Step 2: The last three digits are \(314\), and \(314\) is not divisible by \(8\).
Step 3: Therefore, \(7314\) is not divisible by \(8\).

15) Is \(7290\) divisible by \(9\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(9\).
Step 2: Add the digits: \(7+2+9+0=18\). Since \(18\) is divisible by \(9\), the number passes the rule.
Step 3: Therefore, \(7290\) is divisible by \(9\).

16) Is \(3241\) divisible by \(9\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(9\).
Step 2: Add the digits: \(3+2+4+1=10\). Since \(10\) is not divisible by \(9\), the number fails the rule.
Step 3: Therefore, \(3241\) is not divisible by \(9\).

17) Is \(45870\) divisible by \(10\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(10\).
Step 2: The last digit is \(0\).
Step 3: Therefore, \(45870\) is divisible by \(10\).

18) Is \(45875\) divisible by \(10\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(10\).
Step 2: The last digit is \(5\), not \(0\).
Step 3: Therefore, \(45875\) is not divisible by \(10\).

19) Is \(918082\) divisible by \(11\)? \(\color{green}{Yes}\)

Solution:
Step 1: Use the divisibility rule for \(11\).
Step 2: Use alternating sums: \((9+8+8)-(1+0+2)=25-3=22\). Since \(22\) is divisible by \(11\), the number passes the rule.
Step 3: Therefore, \(918082\) is divisible by \(11\).

20) Is \(123456\) divisible by \(11\)? \(\color{red}{No}\)

Solution:
Step 1: Use the divisibility rule for \(11\).
Step 2: Use alternating sums: \((1+3+5)-(2+4+6)=9-12=-3\). Since \(-3\) is not divisible by \(11\), the number fails the rule.
Step 3: Therefore, \(123456\) is not divisible by \(11\).

What are the Divisibility Rules