3. The number 567xy is completely divisible by 30. The possible of x and y can be a. 2 and 0 b. 1 and 0 c. 0 and 0d. 0 and 1 e. None of these

To determine the values of \( x \) and \( y \) for the number \( 567xy \) that make it divisible by 30, we need to consider the divisibility rules for 30. A number is divisible by 30 if it is divisible by both 2 and 3.

### Step 1: Divisibility by 2
For a number to be divisible by 2, its last digit (i.e., \( y \)) must be even. So, \( y \) must be either 0, 2, 4, 6, or 8.

### Step 2: Divisibility by 3
For a number to be divisible by 3, the sum of its digits must be divisible by 3. The sum of the digits of \( 567xy \) is:
\[
5 + 6 + 7 + x + y = 18 + x + y
\]
For \( 18 + x + y \) to be divisible by 3, the sum \( x + y \) must also be divisible by 3, since 18 is already divisible by 3.

### Step 3: Checking the given options
Let's check the options provided:

- **Option a: \( x = 2 \), \( y = 0 \)**
  \[
  x + y = 2 + 0 = 2 \quad \text{(not divisible by 3)}
  \]
  So, this option is not valid.

- **Option b: \( x = 1 \), \( y = 0 \)**
  \[
  x + y = 1 + 0 = 1 \quad \text{(not divisible by 3)}
  \]
  So, this option is not valid.

- **Option c: \( x = 0 \), \( y = 0 \)**
  \[
  x + y = 0 + 0 = 0 \quad \text{(divisible by 3)}
  \]
  Since \( y = 0 \) is also even, this option is valid.

- **Option d: \( x = 0 \), \( y = 1 \)**
  \[
  x + y = 0 + 1 = 1 \quad \text{(not divisible by 3)}
  \]
  So, this option is not valid.

### Conclusion:
The correct answer is **Option c: \( x = 0 \), \( y = 0 \)**.

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Would you like any further explanation or details?

Here are 5 related questions to deepen your understanding:
1. What is the divisibility rule for 6, and how does it differ from 30?
2. Can a number divisible by 30 have a last digit other than 0?
3. How do you check if a number is divisible by 5?
4. If a number is divisible by both 4 and 5, what is the smallest number it could be divisible by?
5. What happens to divisibility when you add or subtract digits?

**Tip**: Always start with the divisibility rules when solving problems involving factors like 30, 60, or 120, as they break down into smaller prime factors like 2, 3, and 5.

3. The number 567xy is completely divisible by 30. The possible values of x and y can be: a. 2 and 0 b. 1 and 0 c. 0 and 0 d. 0 and 1 e. None of these

Find out how to determine the possible values of x and y for the number 567xy to be divisible by 30. This problem covers divisibility rules for 2 and 3, with a step-by-step solution suitable for Grades 5-7.

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