To solve the problem of finding the total possible numbers between 1 to 100 that give a remainder of 3 when divided by 7, we can follow these steps:
### Step 1: Understand the condition
When a number \( n \) is divided by 7 and gives a remainder of 3, we can express this mathematically as:
\[ n = 7k + 3 \]
where \( k \) is a non-negative integer.
### Step 2: Set the range
We need to find values of \( n \) such that:
\[ 1 \leq n \leq 100 \]
### Step 3: Substitute the expression for \( n \)
Substituting the expression from Step 1 into the inequality:
\[ 1 \leq 7k + 3 \leq 100 \]
### Step 4: Solve the inequalities
First, we solve the left side of the inequality:
\[ 7k + 3 \geq 1 \]
Subtracting 3 from both sides gives:
\[ 7k \geq -2 \]
Since \( k \) is a non-negative integer, the smallest value for \( k \) is 0.
Now, we solve the right side of the inequality:
\[ 7k + 3 \leq 100 \]
Subtracting 3 from both sides gives:
\[ 7k \leq 97 \]
Dividing by 7 gives:
\[ k \leq \frac{97}{7} \approx 13.857 \]
Since \( k \) must be an integer, the maximum value for \( k \) is 13.
### Step 5: List the possible values of \( k \)
The possible integer values for \( k \) are:
\[ k = 0, 1, 2, \ldots, 13 \]
This gives us a total of:
\[ 13 - 0 + 1 = 14 \text{ values} \]
### Step 6: Calculate the corresponding values of \( n \)
Now, we can calculate the corresponding values of \( n \) for each \( k \):
- For \( k = 0 \): \( n = 7(0) + 3 = 3 \)
- For \( k = 1 \): \( n = 7(1) + 3 = 10 \)
- For \( k = 2 \): \( n = 7(2) + 3 = 17 \)
- For \( k = 3 \): \( n = 7(3) + 3 = 24 \)
- For \( k = 4 \): \( n = 7(4) + 3 = 31 \)
- For \( k = 5 \): \( n = 7(5) + 3 = 38 \)
- For \( k = 6 \): \( n = 7(6) + 3 = 45 \)
- For \( k = 7 \): \( n = 7(7) + 3 = 52 \)
- For \( k = 8 \): \( n = 7(8) + 3 = 59 \)
- For \( k = 9 \): \( n = 7(9) + 3 = 66 \)
- For \( k = 10 \): \( n = 7(10) + 3 = 73 \)
- For \( k = 11 \): \( n = 7(11) + 3 = 80 \)
- For \( k = 12 \): \( n = 7(12) + 3 = 87 \)
- For \( k = 13 \): \( n = 7(13) + 3 = 94 \)
### Conclusion
The total possible numbers between 1 and 100 that give a remainder of 3 when divided by 7 are:
\[ 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94 \]
Thus, there are **14 such numbers**.
