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?

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Step-by-step video solution

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00:05 Is every number divisible by three, also divisible by nine?

00:10 Let's take a look at a number that can be divided by three.

00:14 We'll see if this number can also be divided by nine.

00:18 Notice, it doesn't divide evenly by nine.

00:21 So not every number divisible by three is divisible by nine.

00:27 And that's how we solve this question!

Step-by-step written solution

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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

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Will a number divisible by 6 necessarily be divisible by 3?

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!

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