Divisibility Test: Is 684 Divisible by 6?

Q: Why can't I just divide 684 by 6 to check?

You absolutely can, but using divisibility rules is faster and helps avoid calculation errors! Plus, it builds number sense and pattern recognition skills.

Q: What if the number ends in 0? Is it automatically divisible by 6?

Not always! Numbers ending in 0 are divisible by 2, but you still need to check the digit sum rule for 3. For example, 10 ends in 0 but $ 1+0=1 $ isn't divisible by 3.

Q: Do I need to memorize all divisibility rules?

Focus on the most useful ones first: 2, 3, 5, 6, 9, and 10. Once you master these, the patterns become easier to remember and apply!

Q: What if the digit sum is a large number like 27?

Keep going! Add the digits of 27: $ 2+7=9 $. Since 9 is divisible by 3, the original number is too. You can repeat this process until you get a single digit.

Divisibility Rules with Composite Numbers

Determine if the following number is divisible by 6:

$684$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the number is divisible by 6

00:03 According to the ones digit, we know the number is divisible by 2

00:09 Now let's sum the digits and see if the sum is divisible by 3

00:20 The sum is divisible, therefore the number is divisible by 6

00:25 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

Determine if the following number is divisible by 6:

$684$

Step-by-step solution

To determine if the number 684 is divisible by 6, we will apply the relevant divisibility rules:

Divisibility by 2: A number is divisible by 2 if its last digit is even. The last digit of 684 is 4, which is even. Therefore, 684 is divisible by 2.
Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 684 is $6 + 8 + 4 = 18$ . Since 18 is divisible by 3 (because $18 \div 3 = 6$ ), 684 is divisible by 3.

Since 684 is divisible by both 2 and 3, it is divisible by 6.

Therefore, the solution to the problem is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Composite Rule: For 6, check divisibility by both 2 and 3
Technique: Check last digit (4 is even) and digit sum $6+8+4=18$
Verification: Both conditions met means 684 ÷ 6 = 114 exactly ✓

Common Mistakes

Avoid these frequent errors

Testing divisibility by 6 directly
Don't try to divide 684 ÷ 6 immediately = harder mental math and possible errors! This skips the efficient rules and wastes time. Always use the factor rules: check divisibility by 2 AND 3 first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just divide 684 by 6 to check?

You absolutely can, but using divisibility rules is faster and helps avoid calculation errors! Plus, it builds number sense and pattern recognition skills.

What if the number ends in 0? Is it automatically divisible by 6?

Not always! Numbers ending in 0 are divisible by 2, but you still need to check the digit sum rule for 3. For example, 10 ends in 0 but $1+0=1$ isn't divisible by 3.

Do I need to memorize all divisibility rules?

Focus on the most useful ones first: 2, 3, 5, 6, 9, and 10. Once you master these, the patterns become easier to remember and apply!

What if the digit sum is a large number like 27?

Keep going! Add the digits of 27: $2+7=9$ . Since 9 is divisible by 3, the original number is too. You can repeat this process until you get a single digit.

Why does the digit sum rule work for 3?

It's based on how our number system works! When you write 684, you're really writing $6×100 + 8×10 + 4×1$ . The mathematical proof involves modular arithmetic, but the rule always works.

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