Finding The Sum Of Two-Digit Numbers Not Divisible By 3

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Introduction

In the realm of mathematics, there are often intriguing problems that require a blend of arithmetic and logical reasoning to solve. One such problem involves finding the sum of all two-digit natural numbers that are not divisible by 3. This task may seem straightforward at first glance, but it necessitates a systematic approach to avoid errors and arrive at the correct solution. In this comprehensive guide, we will delve into the step-by-step process of solving this problem, providing clear explanations and examples along the way. By understanding the underlying principles and techniques, you will not only be able to solve this specific problem but also gain valuable problem-solving skills applicable to a wide range of mathematical challenges. Let's embark on this mathematical journey together and unravel the solution to this fascinating problem.

Understanding the Problem

To effectively find the sum of two-digit numbers not divisible by 3, it's crucial to first comprehend the scope of the problem. We are looking for all natural numbers between 10 and 99 (inclusive) that do not yield a whole number when divided by 3. These numbers form a subset of the set of all two-digit numbers, and our goal is to determine the sum of this particular subset. The problem requires us to identify these numbers and then apply the principles of arithmetic series to calculate their total sum. Before we jump into the calculations, let's take a moment to explore the concepts and tools that will help us solve this problem efficiently.

Key Concepts

Before we dive into the solution, let's brush up on some key concepts that will be helpful in tackling this problem:

  • Natural Numbers: These are positive integers (1, 2, 3, ...). In our case, we are concerned with natural numbers between 10 and 99.
  • Divisibility: A number is divisible by 3 if the remainder is 0 when divided by 3.
  • Arithmetic Series: A sequence of numbers with a constant difference between consecutive terms. The sum of an arithmetic series can be calculated using a specific formula.

Problem-Solving Strategy

Our strategy to solve this problem involves the following steps:

  1. Identify all two-digit natural numbers.
  2. Determine the two-digit numbers that are divisible by 3.
  3. Calculate the sum of all two-digit numbers.
  4. Calculate the sum of two-digit numbers divisible by 3.
  5. Subtract the sum of numbers divisible by 3 from the sum of all two-digit numbers to obtain the sum of numbers not divisible by 3.

Now that we have a clear understanding of the problem and our approach, let's move on to the step-by-step solution.

Step-by-Step Solution

Now, let's break down the solution into manageable steps, ensuring a clear and logical progression towards the final answer. This methodical approach will not only help us solve the problem accurately but also enhance our problem-solving skills in general. Each step will be explained in detail, with examples and intermediate calculations provided to aid understanding. By following this structured process, we can confidently navigate through the complexities of the problem and arrive at the correct solution. Let's begin by identifying the two-digit natural numbers and then move on to determining those that are divisible by 3.

Step 1: Identify All Two-Digit Natural Numbers

The first step is to identify the range of numbers we are dealing with. Two-digit natural numbers start from 10 and go up to 99. So, we have a sequence of numbers: 10, 11, 12, ..., 99. This sequence forms an arithmetic progression with a common difference of 1. To find the sum of all these numbers, we can use the formula for the sum of an arithmetic series.

Step 2: Determine Two-Digit Numbers Divisible by 3

Next, we need to identify the two-digit numbers that are divisible by 3. The first two-digit number divisible by 3 is 12 (3 x 4), and the last one is 99 (3 x 33). So, the sequence of numbers divisible by 3 is: 12, 15, 18, ..., 99. This is also an arithmetic progression, but with a common difference of 3.

Step 3: Calculate the Sum of All Two-Digit Numbers

To calculate the sum of all two-digit numbers (10 to 99), we use the arithmetic series sum formula:

S = (n/2) * (first term + last term)

where:

  • S is the sum of the series
  • n is the number of terms

First, we need to find the number of terms (n). The number of terms in the sequence 10, 11, ..., 99 is 99 - 10 + 1 = 90.

Now, we can calculate the sum:

S = (90/2) * (10 + 99)

S = 45 * 109

S = 4905

So, the sum of all two-digit numbers is 4905.

Step 4: Calculate the Sum of Two-Digit Numbers Divisible by 3

Now, let's calculate the sum of two-digit numbers divisible by 3 (12, 15, ..., 99). We use the same arithmetic series sum formula.

First, we need to find the number of terms (n) in this sequence. The numbers are multiples of 3, so we can divide each term by 3 to get the sequence 4, 5, ..., 33. The number of terms in this sequence is 33 - 4 + 1 = 30.

Now, we can calculate the sum:

S = (30/2) * (12 + 99)

S = 15 * 111

S = 1665

So, the sum of two-digit numbers divisible by 3 is 1665.

Step 5: Subtract to Find the Sum of Numbers Not Divisible by 3

Finally, to find the sum of two-digit numbers not divisible by 3, we subtract the sum of numbers divisible by 3 from the sum of all two-digit numbers:

Sum of numbers not divisible by 3 = Sum of all two-digit numbers - Sum of numbers divisible by 3

Sum of numbers not divisible by 3 = 4905 - 1665

Sum of numbers not divisible by 3 = 3240

Therefore, the sum of all two-digit natural numbers which are not divisible by 3 is 3240.

Alternative Approach: Direct Calculation

While the previous method involved subtraction, let's explore a more direct approach to calculate the sum. This method reinforces our understanding of arithmetic series and provides an alternative way to solve the problem. By understanding both methods, we can choose the one that best suits our needs and preferences. This direct calculation involves identifying the two arithmetic series formed by the numbers not divisible by 3 and then summing them separately. Let's delve into the details of this alternative approach.

Identifying the Series

The two-digit numbers not divisible by 3 can be divided into two arithmetic series:

  1. Numbers that leave a remainder of 1 when divided by 3: 10, 13, 16, ..., 97
  2. Numbers that leave a remainder of 2 when divided by 3: 11, 14, 17, ..., 98

We will calculate the sum of each series separately and then add them together to get the final answer.

Calculating the Sum of the First Series

For the first series (10, 13, 16, ..., 97), the first term is 10, the common difference is 3, and the last term is 97. To find the number of terms (n), we use the formula:

last term = first term + (n - 1) * common difference

97 = 10 + (n - 1) * 3

87 = (n - 1) * 3

29 = n - 1

n = 30

Now, we can calculate the sum using the arithmetic series sum formula:

S = (n/2) * (first term + last term)

S = (30/2) * (10 + 97)

S = 15 * 107

S = 1605

Calculating the Sum of the Second Series

For the second series (11, 14, 17, ..., 98), the first term is 11, the common difference is 3, and the last term is 98. To find the number of terms (n), we use the formula:

last term = first term + (n - 1) * common difference

98 = 11 + (n - 1) * 3

87 = (n - 1) * 3

29 = n - 1

n = 30

Now, we can calculate the sum using the arithmetic series sum formula:

S = (n/2) * (first term + last term)

S = (30/2) * (11 + 98)

S = 15 * 109

S = 1635

Adding the Sums of the Two Series

Finally, we add the sums of the two series to get the total sum of two-digit numbers not divisible by 3:

Total Sum = Sum of first series + Sum of second series

Total Sum = 1605 + 1635

Total Sum = 3240

This direct calculation method confirms our previous result: the sum of all two-digit natural numbers which are not divisible by 3 is 3240.

Conclusion

In conclusion, we have successfully found the sum of all two-digit natural numbers not divisible by 3. We explored two different methods to solve this problem, both of which led us to the same answer: 3240. The first method involved calculating the sum of all two-digit numbers and then subtracting the sum of those divisible by 3. The second method involved directly calculating the sums of the two arithmetic series formed by numbers that leave remainders of 1 and 2 when divided by 3. Both approaches demonstrate the power of arithmetic series and problem-solving techniques in mathematics. By understanding these methods, you can confidently tackle similar problems and enhance your mathematical abilities. Remember, the key to success in mathematics lies in understanding the underlying concepts and applying them systematically. We hope this comprehensive guide has been helpful in your mathematical journey.

Practice Problems

To solidify your understanding, here are a few practice problems:

  1. Find the sum of all two-digit natural numbers which are not divisible by 5.
  2. Find the sum of all three-digit natural numbers which are divisible by 4.
  3. Find the sum of all natural numbers between 100 and 200 which are not divisible by 7.

Solving these problems will help you reinforce the concepts and techniques discussed in this article. Good luck!