Problem 43 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the sum of integers from 1 to 100 which are divisible by 2 or 5 .

Step-by-Step Solution

Answer

The sum of integers from 1 to 100 which are divisible by 2 or 5 is 3050.

Step 1: Identify the numbers divisible by 2

In this step, we will list down all the numbers between 1 to 100 which are divisible by 2. That is, these numbers will be the even numbers. To find them, the sequence will be: \(2, 4, 6, 8,..., 100\)

Step 2: Identify the numbers divisible by 5

Similarly, in this step, we will list down all the numbers between 1 to 100 which are divisible by 5. They will be the multiples of 5. To find them, the sequence will be: \(5, 10, 15, 20,..., 100\)

Step 3: Sum the numbers divisible by 2

Now we will sum up all the even numbers using the arithmetic progression formula: \[S_n = \frac{n(a_1 + a_n)}{2}\] Here, \(n\) denotes the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term in the sequence. The total number of even numbers from 1 to 100 is 50. So, \(n = 50\), \(a_1 = 2\), \(a_n = 100\). \[S_{even} = \frac{50(2 + 100)}{2} = 25 \times 102 = 2550\]

Step 4: Sum the numbers divisible by 5

We will sum up all the numbers that are multiples of 5 using the arithmetic progression formula again: In this sequence, the total number of multiples of 5 from 1 to 100 is 20. Thus, \(n = 20\), \(a_1 = 5\), \(a_n = 100\). \[S_{five} = \frac{20(5 + 100)}{2} = 10 \times 105 = 1050\]

Step 5: Subtract the numbers divisible by 10 (the common multiples of 2 and 5)

We need to subtract the sum of numbers which are divisible by 10 as they are included twice (once in the even numbers and once in the multiples of 5). Counting the numbers divisible by 10 from 1 to 100, we see that there are 10 such numbers. Thus, \(n = 10\), \(a_1 = 10\), \(a_n = 100\). \[S_{ten} = \frac{10(10 + 100)}{2} = 5 \times 110 = 550\]

Step 6: Calculate the final sum

Now, we will calculate the final sum by adding the sum of numbers divisible by 2 (Step 3) and the sum of numbers divisible by 5 (Step 4), and then subtracting the sum of numbers divisible by 10 (Step 5): \[S_{final} = S_{even} + S_{five} - S_{ten} = 2550 + 1050 - 550 = 3050\] So, the sum of integers from 1 to 100 which are divisible by 2 or 5 is 3050.

Key Concepts

Sum of integers

Summing integers within a specific set involves identifying those integers and performing an addition operation. In this context, we look at integers divisible by certain numbers, like 2 and 5. The process of finding the sum involves:

• Identifying the relevant integers: Determine which numbers meet the criteria of divisibility.
• Using the arithmetic progression formula: This helps simplify adding a series of consecutive numbers.

The formula for the sum of an arithmetic sequence is:\[ S_n = \frac{n(a_1 + a_n)}{2} \]where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
For example, to find the sum of all even numbers from 1 to 100, you translate this into finding the sum of an arithmetic sequence from 2 to 100 with differences of 2.

Divisibility

Divisibility is a core concept in mathematics that describes whether one number can be divided by another without leaving a remainder. In this exercise, you're tasked with finding numbers divisible by 2 or 5.
Numbers divisible by 2 are known as even numbers, which include all multiples of 2 (like 2, 4, 6, etc.).
Similarly, numbers divisible by 5 include all multiples of 5 (like 5, 10, 15, etc.).

• Even numbers inside a range can be systematically identified.
• Multiples of a number like 5 are easily extracted from a set, as they occur at regular intervals.

Understanding divisibility aids in determining which numbers belong to specific arithmetic progressions.

Sequences

In arithmetic progressions, sequences are ordered lists of numbers with a specific pattern or rule. In this exercise, we consider:

• The sequence of even numbers: Begins at 2 and ends at 100, with increments of 2.
• The sequence of multiples of 5: Begins at 5 and ends at 100, with increments of 5.

The formula for an arithmetic sequence allows us to determine important sequence properties, such as the number of terms or the total sum. Understanding sequences is foundational in tackling sums over a range of integer values, helping easily identify the number set to work with.

Common multiples

Common multiples are numbers divisible by two or more other numbers without leaving a remainder. Here, numbers divisible by both 2 and 5 are common multiples and are actually multiples of 10 (since \(10 = 2 \times 5\)).
These numbers appear in both sequences of numbers divisible by 2 and 5, so they are counted twice.
To avoid this duplication in sum calculations:

• Identify the range of numbers divisible by both divisors.
• Calculate their total sum, treating them as a separate sequence.

This method ensures the common multiples are subtracted once from the total calculated sum, leading to accurate results as shown in the exercise.