Animated Solution for Mathematics - Sequence and Series: The sum of all natural numbers 'n' such that $100<n<200$ and H.C.F. $(91,n)>1$ is :

Unlocking the Hidden Patterns

Imagine you are standing before a vast, orderly grid of numbers, ranging from $100$ to $200$. Your task is to find a specific subset of these numbers—those that share a secret connection with the number $91$.

This isn't just about arithmetic; it is about uncovering the hidden structure within the integers. Let us embark on this journey together.

Decoding the Condition

We are given the condition $\text{H.C.F.}(91,n)>1$. To understand this, we must look at the prime factorization of $91$.

For the highest common factor of $n$ and $91$ to be greater than $1$, $n$ must share at least one of these prime factors. In other words, $n$ must be a multiple of $7$ or a multiple of $13$.

This is the core of our problem: we are looking for the union of two sets, $S_7$ and $S_{13}$, within the interval $[100,200]$.

The Arithmetic Progression

Let us first tackle the multiples of $7$. The first multiple of $7$ strictly greater than $100$ is $15\times 7=105$.

The largest multiple of $7$ less than or equal to $200$ is $28\times 7=196$. We have an arithmetic progression: $105,112,\dots ,196$.

The number of terms $k_7$ is calculated as:

The sum $S_7$ is then:

Next, we turn to the multiples of $13$. The first multiple greater than $100$ is $8\times 13=104$, and the largest multiple below $200$ is $15\times 13=195$.

The number of terms $k_{13}$ is found via:

The sum $S_{13}$ is:

The Trap of Double Counting

If we simply add $S_7$ and $S_{13}$, we count the numbers that are multiples of both $7$ and $13$ twice. These are the multiples of $\text{L.C.M.}(7,13)=91$.

In our range $[100,200]$, the only multiple of $91$ is $91\times 2=182$. Thus, the sum of our intersection $S_{91}$ is simply $182$.

To find the total sum, we apply the Principle of Inclusion-Exclusion:

Substituting our values, we get:

This simplifies to $3303-182$, which brings us to our final, elegant result: $3121$. You have successfully navigated the traps and uncovered the sum.