How many numbers lie between 10 and 300, which leaves always a remainder 3 when divided by 4 ?

Question

Accepted Answer

To determine how many numbers lie between 10 and 300 that leave a remainder of 3 when divided by 4, we need to identify the sequence of numbers that satisfy this condition.

### Explanation:

1. **Understanding the Condition:**
   - A number leaves a remainder of 3 when divided by 4 if it can be expressed in the form $4k + 3$, where $k$ is an integer.

2. **Finding the Range:**
   - We need numbers between 10 and 300. Therefore, we need to find the smallest and largest numbers in this range that fit the form $4k + 3$.

3. **Finding the Smallest Number:**
   - The smallest number greater than 10 that leaves a remainder of 3 when divided by 4 is 11. This is because $11 \div 4 = 2$ remainder 3.

4. **Finding the Largest Number:**
   - The largest number less than or equal to 300 that leaves a remainder of 3 when divided by 4 is 299. This is because $299 \div 4 = 74$ remainder 3.

5. **Formulating the Sequence:**
   - The sequence of numbers that satisfy the condition is: 11, 15, 19, ..., 299. This sequence is an arithmetic sequence where the first term $a = 11$ and the common difference $d = 4$.

6. **Finding the Number of Terms:**
   - The general formula for the $n$-th term of an arithmetic sequence is given by:
     $$
     a_n = a + (n-1) \times d
     $$
   - Setting $a_n = 299$, we solve for $n$:
     $$
     299 = 11 + (n-1) \times 4
     $$
     $$
     299 = 11 + 4n - 4
     $$
     $$
     299 = 7 + 4n
     $$
     $$
     292 = 4n
     $$
     $$
     n = \frac{292}{4} = 73
     $$

Thus, there are 73 numbers between 10 and 300 that leave a remainder of 3 when divided by 4.

### Justification for Other Options:

- **Option 2 (71):** This option is incorrect because, as calculated, the number of terms in the sequence is 73, not 71.
- **Option 3 (68):** This option is incorrect because it underestimates the number of terms in the sequence.
- **Option 4 (72):** This option is incorrect because it also underestimates the number of terms in the sequence.

Therefore, the correct answer is Option 1, which is 73.