Divisibility Test: Is 685 Divisible by 9?

Q: Why does adding digits tell me if a number is divisible by 9?

This works because of how our base-10 number system relates to multiples of 9. When you add digits, you're essentially finding the remainder when divided by 9 - it's a mathematical shortcut that always works!

Q: Does this trick work for other numbers besides 9?

The digit sum method works perfectly for 3 and 9 only. For other numbers like 2, 4, 5, 6, 8, or 10, you need different divisibility rules. Each number has its own special pattern!

Q: What if I make an addition mistake when finding the digit sum?

Double-check your addition! For 685: $ 6 + 8 = 14 $, then $ 14 + 5 = 19 $. Write it step-by-step to avoid errors, especially with larger numbers.

Q: Can a number be divisible by 9 but not by 3?

No, never! If a number is divisible by 9, it's automatically divisible by 3 too, since 9 = 3 × 3. But a number can be divisible by 3 without being divisible by 9.

Divisibility Rules with Digit Sum Method

Is the number below divisible by 9?

$685$

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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 9

00:03 A number is divisible by 9 only if the sum of its digits is divisible by 9

00:07 Let's sum its digits and divide by 9

00:11 The sum of digits is not divisible by 9, therefore the number is not divisible by 9

00:17 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

Is the number below divisible by 9?

$685$

Step-by-step solution

To determine whether the number $685$ is divisible by $9$ , we use the divisibility rule for $9$ : a number is divisible by $9$ if the sum of its digits is divisible by $9$ .

Let's calculate the sum of the digits in $685$ :

$6 + 8 + 5 = 19$

Now, we check if $19$ is divisible by $9$ . In this case, $19 \div 9 = 2$ with a remainder of $1$ .

Since $19$ is not divisible by $9$ , the number $685$ is also not divisible by $9$ .

Therefore, the answer to the problem is that $685$ is not divisible by $9$ . Hence, the correct choice is:

: No

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: A number is divisible by 9 if digit sum is divisible by 9
Technique: Add all digits: $6 + 8 + 5 = 19$ , then check if 19 ÷ 9 has no remainder
Check: Since $19 \div 9 = 2$ remainder 1, the original number 685 is not divisible by 9 ✓

Common Mistakes

Avoid these frequent errors

Dividing the original number instead of using digit sum
Don't calculate 685 ÷ 9 directly = complicated long division! This wastes time and increases error chances. Always use the digit sum shortcut: add the digits first, then check if that sum divides by 9.

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 does adding digits tell me if a number is divisible by 9?

This works because of how our base-10 number system relates to multiples of 9. When you add digits, you're essentially finding the remainder when divided by 9 - it's a mathematical shortcut that always works!

What if the digit sum is still a large number?

Keep adding the digits! For example, if your sum is 47, add $4 + 7 = 11$ . Since 11 isn't divisible by 9, the original number isn't either. Keep going until you get a single digit.

Does this trick work for other numbers besides 9?

The digit sum method works perfectly for 3 and 9 only. For other numbers like 2, 4, 5, 6, 8, or 10, you need different divisibility rules. Each number has its own special pattern!

What if I make an addition mistake when finding the digit sum?

Double-check your addition! For 685: $6 + 8 = 14$ , then $14 + 5 = 19$ . Write it step-by-step to avoid errors, especially with larger numbers.

Can a number be divisible by 9 but not by 3?

No, never! If a number is divisible by 9, it's automatically divisible by 3 too, since 9 = 3 × 3. But a number can be divisible by 3 without being divisible by 9.

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