What is the sum of the natural numbers from l to 100, which are not divisible by 7 and 11?2946

To find the sum of the natural numbers from $1$ to $100$ that are not divisible by $7$ or $11$, we will follow the outlined steps.

1. Break Down the Problem

We need to:

1. Calculate the total sum of natural numbers from $1$ to $100$.
2. Calculate the sum of numbers from $1$ to $100$ that are divisible by $7$.
3. Calculate the sum of numbers from $1$ to $100$ that are divisible by $11$.
4. Calculate the sum of numbers from $1$ to $100$ that are divisible by both $7$ and $11$ (i.e., divisible by $77$).
5. Use the inclusion-exclusion principle to find the sum of numbers from $1$ to $100$ that are divisible by $7$ or $11$.
6. Subtract the above sum from the total sum to get the sum of numbers not divisible by $7$ or $11$.

2. Relevant Concepts

• The sum of the first $n$ natural numbers can be calculated using the formula: $S_n = \frac{n(n + 1)}{2}$
• For the sum of multiples of $k$ up to $n$: $S = k \cdot \left(\frac{m(m + 1)}{2}\right)$ where $m = \left\lfloor \frac{n}{k} \right\rfloor$.

3. Analysis and Detail

1. Total Sum from $1$ to $100$: $S_{100} = \frac{100(100 + 1)}{2} = 5050$

2. Sum of numbers divisible by $7$:

   • Number of terms: $m = \left\lfloor \frac{100}{7} \right\rfloor = 14$
   • Sum of multiples of $7$: $S_7 = 7 \cdot \frac{14(14 + 1)}{2} = 7 \cdot \frac{14 \cdot 15}{2} = 7 \cdot 105 = 735$

3. Sum of numbers divisible by $11$:

   • Number of terms: $m = \left\lfloor \frac{100}{11} \right\rfloor = 9$
   • Sum of multiples of $11$: $S_{11} = 11 \cdot \frac{9(9 + 1)}{2} = 11 \cdot \frac{9 \cdot 10}{2} = 11 \cdot 45 = 495$

4. Sum of numbers divisible by $77$ (LCM of $7$ and $11$):

   • Number of terms: $m = \left\lfloor \frac{100}{77} \right\rfloor = 1$
   • Sum of multiples of $77$: $S_{77} = 77 \cdot \frac{1(1 + 1)}{2} = 77 \cdot 1 = 77$

5. Using inclusion-exclusion: $S_{7 \cup 11} = S_7 + S_{11} - S_{77} = 735 + 495 - 77 = 1153$

6. Final Calculation: $\text{Sum of numbers not divisible by \(7\) or \(11\)} = S_{100} - S_{7 \cup 11} = 5050 - 1153 = 3897$

4. Verify and Summarize

After careful calculations, we find that the sum of natural numbers from $1$ to $100$ that are not divisible by $7$ and $11$ is $3897$.

Final Answer

The sum of the natural numbers from $1$ to $100$ that are not divisible by $7$ and $11$ is 3897.