Divisibility Rules for 3, 6 and 9: Test for divisibility

To determine if the number 564 is divisible by 3, we apply the divisibility rule for 3:

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

Let's calculate the sum of the digits of 564:

$5 + 6 + 4 = 15$

Next, we check if 15 is divisible by 3. Since 15 can be divided by 3 without a remainder, it is divisible by 3:

$15 \div 3 = 5$

Therefore, based on the divisibility rule, 564 is divisible by 3.

Thus, the correct answer is Yes.

To determine if 673 is divisible by 3, we must use the divisibility rule for 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3.

First, we'll calculate the sum of the digits: $6 + 7 + 3$.

Calculating this, we get: $6 + 7 + 3 = 16$.

Next, we check if 16 is divisible by 3. Dividing 16 by 3 gives a quotient of 5 and a remainder of 1.

Since 16 is not divisible by 3 (as it leaves a remainder), we conclude that 673 is not divisible by 3.

Thus, the correct answer is No.

To determine if 352 is divisible by 3, we need to follow these steps:

• Calculate the sum of its digits.
• Check if this sum is divisible by 3.

Let's work through the procedure:

The number $352$ consists of the digits 3, 5, and 2.

Step 1: Calculate the sum of the digits.

The sum is $3 + 5 + 2 = 10$.

Step 2: Check if 10 is divisible by 3.

Since 10 divided by 3 gives a remainder, 10 is not divisible by 3.

Therefore, the number 352 is not divisible by 3.

The correct answer is No.

To determine if the number $132$ is divisible by $3$, we can apply the rule for divisibility by $3$, which involves summing the digits of the number.

Step-by-step solution:

• Step 1: Sum the digits of the number $132$.
  $1 + 3 + 2 = 6$
• Step 2: Check if the sum is divisible by $3$.
  $6 \div 3 = 2$, which is an integer.

Since the sum of the digits is $6$ and $6$ is divisible by $3$, the number $132$ is also divisible by $3$.

Therefore, the number $132$ is divisible by $3$, and the correct choice is:

Yes

To determine if the number 999 is divisible by 9, we will apply the divisibility rule for 9. According to this rule, a number is divisible by 9 if the sum of its digits is divisible by 9.

Let's work through the process:

• Step 1: Identify the digits of the number 999. They are 9, 9, and 9.
• Step 2: Calculate the sum of these digits: $9 + 9 + 9 = 27$.
• Step 3: Check if the sum (27) is divisible by 9. Since $27 \div 9 = 3$, and 27 is exactly divisible by 9, the original number 999 is also divisible by 9.

Therefore, according to the divisibility rule, the number 999 is divisible by 9.

The correct answer is Yes.

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

To determine if 987 is divisible by 9, we use the divisibility rule that a number is divisible by 9 if the sum of its digits is also divisible by 9.

Let's follow these steps:

• Step 1: Find the sum of the digits of 987.

We have the digits 9, 8, and 7. Therefore, their sum is:

$9 + 8 + 7 = 24$

• Step 2: Check if the sum, 24, is divisible by 9.

We divide 24 by 9:

$24 \div 9 \approx 2.67$

Since 24 does not evenly divide by 9 (it does not result in an integer), 24 is not divisible by 9.

Thus, since the sum of the digits (24) is not divisible by 9, the number 987 is not divisible by 9.

Therefore, the solution to the problem is No.

To determine if 189 is divisible by 9, we apply the divisibility rule for 9:

• Step 1: Calculate the sum of the digits of 189.

The number 189 can be broken down into its digits: 1, 8, and 9. We find the sum of these digits:

$1 + 8 + 9 = 18$

• Step 2: Check if the sum, 18, is divisible by 9.

We know that 18 divided by 9 equals 2, which is a whole number, meaning 18 is divisible by 9.

Since the sum of the digits (18) is divisible by 9, it follows that 189 itself is divisible by 9.

Therefore, the number 189 is divisible by 9.

Final Answer: Yes

To determine if the number 484 is divisible by 6, we must confirm that it satisfies both divisibility rules for 2 and 3:

• Divisibility by 2: A number is divisible by 2 if its last digit is even. The last digit of 484 is 4, which is even. Thus, 484 is divisible by 2.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For 484, the sum of the digits is $4 + 8 + 4 = 16$. To check divisibility by 3, we evaluate whether 16 is divisible by 3, which it is not.

Since 484 is not divisible by 3, even though it is divisible by 2, it cannot be divisible by 6. Therefore, 484 is not divisible by 6.

In conclusion, the answer to the problem is: No.

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.

To determine if 681 is divisible by 6, we need to apply the divisibility rules for both 2 and 3:

• Divisibility by 2: A number is divisible by 2 if its last digit is even. The last digit of 681 is 1, which is odd. Therefore, 681 is not 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 681 is $6 + 8 + 1 = 15$. Since 15 is divisible by 3, 681 meets this condition.

Since 681 is not divisible by 2, even though it is divisible by 3, we conclude that 681 is not divisible by 6.

Therefore, the solution to the problem is No.

To solve this problem, we'll follow these steps:

• Step 1: Check divisibility by 2.
• Step 2: Check divisibility by 3.
• Step 3: Conclude divisibility by 6 based on Steps 1 and 2.

Now, let's work through each step:

Step 1: Check divisibility by 2.
The last digit of 876 is 6, which is an even number. Thus, 876 is divisible by 2.

Step 2: Check divisibility by 3.
The sum of the digits is $8 + 7 + 6 = 21$.
Check if 21 is divisible by 3: divide $21 \div 3 = 7$, which is a whole number. Thus, 21 is divisible by 3, so 876 is also divisible by 3.

Step 3: Conclude divisibility by 6.
Since 876 is divisible by both 2 and 3, it is also divisible by 6.

Therefore, the number 876 is divisible by 6, and the answer to the problem is Yes.