A number is divisible by if the sum of its digits is a multiple of .
A number is divisible by if it is even and also a multiple of .
A number is divisible by if the sum of its digits is a multiple of .
Determine if the following number is divisible by 3:
\( 564 \)
Determine if the following number is divisible by 3:
\( 673 \)
Determine if the following number is divisible by 3:
\( 352 \)
Determine if the following number is divisible by 3:
\( 132 \)
Will a number divisible by 6 necessarily be divisible by 2?
Determine if the following number is divisible by 3:
To determine if the number 564 is divisible by 3, we apply the divisibility rule for 3:
Let's calculate the sum of the digits of 564:
Next, we check if 15 is divisible by 3. Since 15 can be divided by 3 without a remainder, it is divisible by 3:
Therefore, based on the divisibility rule, 564 is divisible by 3.
Thus, the correct answer is Yes.
Yes
Determine if the following number is divisible by 3:
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: .
Calculating this, we get: .
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.
No
Determine if the following number is divisible by 3:
To determine if 352 is divisible by 3, we need to follow these steps:
Let's work through the procedure:
The number consists of the digits 3, 5, and 2.
Step 1: Calculate the sum of the digits.
The sum is .
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.
No
Determine if the following number is divisible by 3:
To determine if the number is divisible by , we can apply the rule for divisibility by , which involves summing the digits of the number.
Step-by-step solution:
Since the sum of the digits is and is divisible by , the number is also divisible by .
Therefore, the number is divisible by , and the correct choice is:
Yes
Yes
Will a number divisible by 6 necessarily be divisible by 2?
In order to determine if a number divisible by 6 is also divisible by 2, we first review the divisibility rules:
Consider a number that is divisible by 6. By definition, since 6 itself factors into 2 multiplied by 3, any number divisible by 6 must be divisible by 2 and 3. This means that any number divisible by 6 is automatically divisible by 2 because 2 is a part of its factorization.
Therefore, yes, any number divisible by 6 will necessarily be divisible by 2 as per the rule of divisibility.
Thus, the correct choice is:
This conclusion adheres strictly to divisibility rules and confirms the assertion that being divisible by 6 includes being divisible by 2.
Yes
Will a number divisible by 6 necessarily be divisible by 3?
Is the number below divisible by 9?
\( 999 \)
Is the number below divisible by 9?
\( 685 \)
Is the number below divisible by 9?
\( 987 \)
Is the number below divisible by 9?
\( 189 \)
Will a number divisible by 6 necessarily be divisible by 3?
To determine whether a number divisible by 6 is necessarily divisible by 3, we need to understand the properties of divisibility for the numbers involved.
Let's analyze the problem step by step:
Therefore, the answer to the problem is Yes.
Yes
Is the number below divisible by 9?
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:
Therefore, according to the divisibility rule, the number 999 is divisible by 9.
The correct answer is Yes.
Yes
Is the number below divisible by 9?
To determine whether the number is divisible by , we use the divisibility rule for : a number is divisible by if the sum of its digits is divisible by .
Let's calculate the sum of the digits in :
Now, we check if is divisible by . In this case, with a remainder of .
Since is not divisible by , the number is also not divisible by .
Therefore, the answer to the problem is that is not divisible by . Hence, the correct choice is:
No
Is the number below divisible by 9?
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:
We have the digits 9, 8, and 7. Therefore, their sum is:
We divide 24 by 9:
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.
No
Is the number below divisible by 9?
To determine if 189 is divisible by 9, we apply the divisibility rule for 9:
The number 189 can be broken down into its digits: 1, 8, and 9. We find the sum of these digits:
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
Yes
Will a number divisible by 2 necessarily be divisible by 6?
Will a number divisible by 3 necessarily be divisible by 9?
Will a number divisible by 9 necessarily be divisible by 3?
Will a number divisible by 3 necessarily be divisible by 6?
Will a number divisible by 9 necessarily be divisible by 6?
Will a number divisible by 2 necessarily be divisible by 6?
To determine if a number divisible by 2 is also divisible by 6, we need to understand the rules of divisibility:
Now, let's analyze the implication of these rules:
Since a number divisible by 2 is even, it satisfies the first condition for divisibility by 6. However, it still needs to meet the second condition—divisibility by 3—to be divisible by 6. This implies that not all even numbers (divisible by 2) are multiples of 3.
For example, consider the number 4:
Therefore, understanding the definitions, we can see that a number divisible by 2 is not necessarily divisible by 6. These two criteria must both be met for a number to be divisible by 6. Consequently, the correct answer is "No".
In conclusion, a number being divisible by 2 does not guarantee that it is divisible by 6.
No
Will a number divisible by 3 necessarily be divisible by 9?
To solve this problem, we need to understand the divisibility rules for 3 and 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 , 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 , 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.
No
Will a number divisible by 9 necessarily be divisible by 3?
To solve this problem, we need to apply the divisibility rules for both 9 and 3.
Therefore, it follows that if a number is divisible by 9, it must be divisible by 3, because the divisibility by 9 inherently satisfies the divisibility condition for 3.
Thus, the correct answer is Yes.
Yes
Will a number divisible by 3 necessarily be divisible by 6?
To determine if a number divisible by 3 is necessarily divisible by 6, we must apply the divisibility rules for both 3 and 6:
To explore this question, let's consider a counterexample:
Take the number . The sum of its digits is , which is divisible by 3, so 9 is divisible by 3.
However, 9 is not even, so it is not divisible by 2. As a result, 9 is not divisible by 6 (because it does not satisfy the requirement to be divisible by both 2 and 3).
This counterexample demonstrates that a number divisible by 3 is not necessarily divisible by 6.
Therefore, the statement is incorrect, and the answer is No.
No
Will a number divisible by 9 necessarily be divisible by 6?
To determine if a number divisible by 9 is necessarily divisible by 6, let's explore the divisibility rules.
A number is divisible by 9 if the sum of its digits is divisible by 9. Consequently, such a number is also divisible by 3 since divisibility by 9 implies divisibility by 3.
For divisibility by 6, a number must be divisible by both 2 and 3. We've established that a number divisible by 9 is also divisible by 3, so we now need to check whether it is necessarily divisible by 2.
Consider an example: the number 27 is divisible by 9 since , which is divisible by 9. However, 27 is odd (since ), and thus, not divisible by 2.
Since 27 is not divisible by both 2 and 3, this number is not divisible by 6.
Therefore, a number divisible by 9 is not necessarily divisible by 6. The correct answer is No.
No