A 4 - digit number 7a3b is divisible by both 5 and 9. What could this 4 - digit number be?

To solve the problem of finding the 4-digit number \( 7a3b \) that is divisible by both 5 and 9, we need to consider the following conditions:

1. **Divisibility by 5**: For a number to be divisible by 5, its last digit \( b \) must be 0 or 5.
2. **Divisibility by 9**: For a number to be divisible by 9, the sum of its digits must be a multiple of 9.

### Step 1: Apply the Divisibility Rule for 5
Since \( 7a3b \) must be divisible by 5, \( b \) can only be 0 or 5.

### Step 2: Apply the Divisibility Rule for 9
The sum of the digits in \( 7a3b \) is \( 7 + a + 3 + b \). This sum must be a multiple of 9.

So, we have:
\[ 7 + a + 3 + b = 10 + a + b \]

This sum, \( 10 + a + b \), must be divisible by 9.

### Step 3: Consider Each Case for \( b \)
We consider the two possibilities for \( b \):

1. **If \( b = 0 \)**:
   \[
   10 + a + 0 = 10 + a
   \]
   \( 10 + a \) must be divisible by 9. The possible value for \( 10 + a \) that is a multiple of 9 is 18 (since \( 9 \times 2 = 18 \)). Therefore, \( a = 8 \).

   In this case, the number is \( 7830 \).

2. **If \( b = 5 \)**:
   \[
   10 + a + 5 = 15 + a
   \]
   \( 15 + a \) must be divisible by 9. The possible value for \( 15 + a \) that is a multiple of 9 is 18 (since \( 9 \times 2 = 18 \)). Therefore, \( a = 3 \).

   In this case, the number is \( 7435 \).

### Step 4: Conclusion
The possible 4-digit numbers that meet the conditions are **7830** and **7435**.

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Would you like more details on how these conditions work or have any further questions?

Here are some related questions:
1. How do you check divisibility by other numbers, such as 6 or 12?
2. What if the number was required to be divisible by both 3 and 8?
3. Could there be more than one number if different digits were allowed?
4. Why are \( b = 0 \) and \( b = 5 \) the only possibilities for divisibility by 5?
5. What if \( b \) was restricted to non-zero digits only?

**Tip**: Always check the smallest condition first when working with multiple divisibility rules. This can narrow down possibilities quickly!

Discover how to find a 4-digit number 7a3b that is divisible by both 5 and 9. Learn about divisibility rules and the conditions required for a number to meet these criteria. Suitable for students in Grades 5-7.

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