Divisibility Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Divisibility Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding when one whole number divides evenly into another, leaving no remainder—the foundation of factor and multiple relationships.

Can you share 12 cookies equally among 4 people? Yes, 3 each. 12 is divisible by 4.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Divisibility rules reveal structure: a is divisible by b if a = b \times k for some integer k.

Common stuck point: Learning the shortcut tests (divisible by 3 if digit sum is divisible by 3).

Sense of Study hint: Add up all the digits of the number. If that sum is divisible by 3, the original number is too. Practice similar shortcuts for 2, 5, 9, and 10.

Worked Examples

Example 1

easy
Use divisibility rules to determine whether 4{,}836 is divisible by 2, 3, 4, 6, and 9.

Solution

  1. 1
    By 2: last digit is 6 (even). Yes.
  2. 2
    By 3: digit sum = 4+8+3+6 = 21; 21 \div 3 = 7. Yes.
  3. 3
    By 4: last two digits 36; 36 \div 4 = 9. Yes.
  4. 4
    By 6: divisible by both 2 and 3. Yes.
  5. 5
    By 9: digit sum 21; 21 \div 9 = 2.33\ldots Not a whole number. No.

Answer

4{,}836 is divisible by 2, 3, 4, 6 but not by 9.
Divisibility rules are shortcuts derived from properties of our base-10 system. The rules for 2 and 5 check the last digit; for 3 and 9, sum the digits; for 4, check the last two digits. These avoid long division for quick classification.

Example 2

medium
Explain why the divisibility rule for 3 works: a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Test 7{,}215 for divisibility by 5, 9, and 10 using divisibility rules.

Example 2

medium
A number N leaves remainder 2 when divided by 5 and remainder 1 when divided by 3. What are the possible last digits of N, and is N divisible by 15?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

division