To find the number of 4-digit numbers that satisfy the given condition, we need to find the range of numbers that satisfy the condition and then count how many numbers fall within that range.

We are given that the numbers leave a remainder of 3 when divided by 7 and 9. This means that the numbers can be expressed as:
- Number = 7a + 3
- Number = 9b + 3

where a and b are integers.

To find the range of numbers, we need to find the smallest and largest possible values for a and b.

For a:
Since the number is a 4-digit number, the smallest possible value for a is 1000/7 = 142.85. However, a must be an integer, so the smallest possible value for a is 143.
The largest possible value for a is (9999-3)/7 = 1428.

Similarly, for b:
The smallest possible value for b is 1000/9 = 111.11. But b must be an integer, so the smallest possible value for b is 112.
The largest possible value for b is (9999-3)/9 = 1111.

Hence, the range of values for a is 143 to 1428, and for b is 112 to 1111.

Now, we need to find how many numbers fall within this range. We can do this by finding the number of values for a and b and multiplying them together.

Number of values for a = 1428 - 143 + 1 = 1286
Number of values for b = 1111 - 112 + 1 = 1000

Total number of numbers = Number of values for a * Number of values for b = 1286 * 1000 = 1,286,000

However, we need to find the number of 4-digit numbers, so we need to subtract the number of 3-digit numbers from this count.

Number of 3-digit numbers = Number of values for a when a has 3 digits = 999 - 143 + 1 = 857

Total number of 4-digit numbers = Total number of numbers - Number of 3-digit numbers = 1,286,000 - 857 = 1,285,143

Therefore, the correct answer is option C) 143.