Divisibility Rule: Does Being Divisible by 3 Imply Divisibility by 9?

Q: Why isn't every number divisible by 3 also divisible by 9?

Because 9 is stricter than 3! While 3 only needs the digit sum divisible by 3, 9 needs the digit sum divisible by 9. For example, 12 has digit sum 3, which works for 3 but not for 9.

Q: Can you give me more examples of numbers divisible by 3 but not 9?

Absolutely! Try 6, 15, 21, 24, 30. All have digit sums divisible by 3 but not by 9. For instance: $ 2 + 1 = 3 $ for 21, and 3 ÷ 3 = 1 but 3 ÷ 9 doesn't work!

Q: How do I remember the difference between these rules?

Think of it as levels: divisible by 3 is level 1, divisible by 9 is level 2. You need level 2 to get level 1, but level 1 doesn't guarantee level 2!

Q: What's the easiest way to test this with examples?

Pick simple numbers! Start with 12: $ 1 + 2 = 3 $. Since 3 ÷ 3 = 1 (works) but 3 ÷ 9 doesn't work, you've found your counterexample!

Divisibility Rules with Counterexample Analysis

Will a number divisible by 3 necessarily be divisible by 9?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is every number divisible by 3 also divisible by 9?

00:03 Let's take an example of a number divisible by 3

00:07 We can see that this number is not divisible by 9

00:13 Therefore, not every number divisible by 3 is divisible by 9

00:16 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Will a number divisible by 3 necessarily be divisible by 9?

Step-by-step solution

To solve this problem, we need to understand the divisibility rules for 3 and 9:

A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 9 if the sum of its digits is divisible by 9.

Let's evaluate whether a number divisible by 3 is necessarily divisible by 9:

Consider the number 12. The sum of its digits is $1 + 2 = 3$ , which is divisible by 3, so 12 is divisible by 3. However, when we check divisibility by 9, 12 is not divisible by 9 because 3 is not divisible by 9.

Now consider another number, like 18. The sum of its digits is $1 + 8 = 9$ , which is divisible by both 3 and 9. Thus, 18 is divisible by both.

These examples demonstrate that while some numbers divisible by 3 are also divisible by 9 (e.g., 18), not all are (e.g., 12).

Therefore, a number being divisible by 3 does not necessarily mean it is divisible by 9.

The correct answer is No.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Divisible by 3 means digit sum divisible by 3
Technique: Test with 12: digit sum 3, divisible by 3 but not 9
Check: Find counterexample like 15 where 1+5=6, divisible by 3 not 9 ✓

Common Mistakes

Avoid these frequent errors

Assuming divisibility by 3 guarantees divisibility by 9
Don't assume that if 3 divides a number, then 9 must divide it too = wrong conclusion! This ignores that 9 requires stricter conditions than 3. Always test with specific examples like 12 or 15 to find counterexamples.

Practice Quiz

Test your knowledge with interactive questions

Determine if the following number is divisible by 3:

$$ 352 $$

FAQ

Everything you need to know about this question

Why isn't every number divisible by 3 also divisible by 9?

Because 9 is stricter than 3! While 3 only needs the digit sum divisible by 3, 9 needs the digit sum divisible by 9. For example, 12 has digit sum 3, which works for 3 but not for 9.

Can you give me more examples of numbers divisible by 3 but not 9?

Absolutely! Try 6, 15, 21, 24, 30. All have digit sums divisible by 3 but not by 9. For instance: $2 + 1 = 3$ for 21, and 3 ÷ 3 = 1 but 3 ÷ 9 doesn't work!

When IS a number divisible by both 3 and 9?

When its digit sum is divisible by 9! Since 9 is a multiple of 3, any number divisible by 9 is automatically divisible by 3. Examples: 18, 27, 36, 45.

How do I remember the difference between these rules?

Think of it as levels: divisible by 3 is level 1, divisible by 9 is level 2. You need level 2 to get level 1, but level 1 doesn't guarantee level 2!

What's the easiest way to test this with examples?

Pick simple numbers! Start with 12: $1 + 2 = 3$ . Since 3 ÷ 3 = 1 (works) but 3 ÷ 9 doesn't work, you've found your counterexample!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Division - Advanced questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime