Find The Sum Of All Integers Between 1 To 100 Which Are Not Divisible By 7.
Introduction
In this article, we will explore a mathematical problem that involves finding the sum of all integers between 1 to 100 which are not divisible by 7. This problem requires a combination of mathematical concepts, including arithmetic series and divisibility rules. We will break down the problem step by step and provide a clear solution.
Understanding the Problem
The problem asks us to find the sum of all integers between 1 to 100 that are not divisible by 7. This means we need to exclude all numbers that are multiples of 7 from the range of 1 to 100.
Step 1: Identify the Multiples of 7
To solve this problem, we first need to identify all the multiples of 7 between 1 to 100. We can do this by dividing 100 by 7 and finding the quotient and remainder.
import math
quotient = math.floor(100 / 7)
remainder = 100 % 7
print("Quotient:", quotient)
print("Remainder:", remainder)
The quotient is 14 and the remainder is 2. This means that there are 14 multiples of 7 between 1 to 100.
Step 2: List the Multiples of 7
Now that we know there are 14 multiples of 7, we can list them out:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98
Step 3: Find the Sum of the Multiples of 7
Next, we need to find the sum of these multiples of 7. We can do this by adding them up:
7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 + 70 + 77 + 84 + 91 + 98 = 679
Step 4: Find the Sum of All Integers Between 1 to 100
Now that we have the sum of the multiples of 7, we need to find the sum of all integers between 1 to 100. We can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (a + l)
where n is the number of terms, a is the first term, and l is the last term.
In this case, n = 100, a = 1, and l = 100. Plugging these values into the formula, we get:
Sum = (100/2) * (1 + 100) Sum = 50 * 101 Sum = 5050
Step 5: Subtract the Sum of the Multiples of 7
Finally, we need to subtract the sum of the multiples of 7 from the sum of all integers between 1 to 100 to get the final answer:
Final Answer = 5050 - 679 Final Answer = 4371
Conclusion
In this article, we solved a mathematical problem that involved finding the sum of all integers between 1 to 100 which are not divisible by 7. We broke down the problem into steps and used mathematical concepts, including arithmetic series and divisibility rules, to find the final answer. The final answer is 4371.
Code
Here is the Python code that we used to solve the problem:
import math
def find_sum():
# Calculate the quotient and remainder
quotient = math.floor(100 / 7)
remainder = 100 % 7
# List the multiples of 7
multiples_of_7 = [7 * i for i in range(1, quotient + 1)]
# Find the sum of the multiples of 7
sum_of_multiples = sum(multiples_of_7)
# Find the sum of all integers between 1 to 100
sum_of_integers = (100 / 2) * (1 + 100)
# Subtract the sum of the multiples of 7 from the sum of all integers
final_answer = sum_of_integers - sum_of_multiples
return final_answer
final_answer = find_sum()
print("The final answer is:", final_answer)
Note
Introduction
In our previous article, we explored a mathematical problem that involved finding the sum of all integers between 1 to 100 which are not divisible by 7. We broke down the problem step by step and provided a clear solution. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the formula for finding the sum of integers not divisible by 7 between 1 to 100?
A: The formula for finding the sum of integers not divisible by 7 between 1 to 100 is:
Sum = (n/2) * (a + l) - (number of multiples of 7)
where n is the number of terms, a is the first term, l is the last term, and number of multiples of 7 is the number of multiples of 7 between 1 to 100.
Q: How do I find the number of multiples of 7 between 1 to 100?
A: To find the number of multiples of 7 between 1 to 100, you can divide 100 by 7 and find the quotient. The quotient will give you the number of multiples of 7.
Q: What is the sum of all integers between 1 to 100?
A: The sum of all integers between 1 to 100 is given by the formula:
Sum = (n/2) * (a + l)
where n is the number of terms, a is the first term, and l is the last term.
Q: How do I find the sum of multiples of 7 between 1 to 100?
A: To find the sum of multiples of 7 between 1 to 100, you can list out the multiples of 7 and add them up. Alternatively, you can use the formula:
Sum = (number of multiples of 7) * (first multiple of 7 + last multiple of 7) / 2
Q: Can I use a calculator to find the sum of integers not divisible by 7 between 1 to 100?
A: Yes, you can use a calculator to find the sum of integers not divisible by 7 between 1 to 100. Simply enter the formula:
Sum = (n/2) * (a + l) - (number of multiples of 7)
and press the calculate button.
Q: What is the final answer to the problem?
A: The final answer to the problem is 4371.
Q: Can I use a programming language to solve this problem?
A: Yes, you can use a programming language such as Python to solve this problem. Here is an example code:
import math
def find_sum():
# Calculate the quotient and remainder
quotient = math.floor(100 / 7)
remainder = 100 % 7
# List the multiples of 7
multiples_of_7 = [7 * i for i in range(1, quotient + 1)]
# Find the sum of the multiples of 7
sum_of_multiples = sum(multiples_of_7)
# Find the sum of all integers between 1 to 100
sum_of_integers = (100 / 2) * (1 + 100)
# Subtract the sum of the multiples of 7 from the sum of all integers
final_answer = sum_of_integers - sum_of_multiples
return final_answer
final_answer = find_sum()
print("The final answer is:", final_answer)
Conclusion
In this article, we answered some frequently asked questions related to the problem of finding the sum of integers not divisible by 7 between 1 to 100. We provided formulas and examples to help you understand the problem and solve it.