## Let $d(k)$ be the sum of all divisors of $k$.

We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.

For example, $S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + (6) + d(3) + d(6) + d(9) = 59$.

You are given that $S(10^3) = 563576517282$ and $S(10^5) \bmod 10^9 = 215766508$.

Find $S(10^{11}) \bmod 10^9$.

### To solve this problem, we will first implement a function to calculate the sum of divisors ($d(k)$) for a given number $k$. Then, we will use this function to calculate the value of $S(N)$ for increasing values of $N$ until we reach $N=10^{11}$. Finally, we will take the modulus of the final result $S(10^{11})$ with $10^9$ to obtain the answer.

Let’s start by implementing the function to calculate the sum of divisors:

“`python

def sum_of_divisors(k):

divisors_sum = 0

for i in range(1, int(k**0.5) + 1):

if k % i == 0:

divisors_sum += i

if k // i != i:

divisors_sum += k // i

return divisors_sum

“`

This function works by iterating over all numbers from 1 to the square root of `k`. If the current number is a divisor of `k`, it adds the current divisor to the sum (`divisors_sum`) and also adds the corresponding divisor (`k // i`) to the sum if it is different from the current divisor.

Now, let’s implement the function to calculate the value of $S(N)$:

“`python

def calculate_S(N):

result = 0

for i in range(1, N + 1):

for j in range(1, N + 1):

result += sum_of_divisors(i * j)

return result

“`

This function iterates over all values of `i` from 1 to `N` and `j` from 1 to `N`, and for each pair of `i` and `j`, it calculates `d(i * j)` using the `sum_of_divisors` function and adds it to the result.

Finally, we can calculate the value of $S(10^{11}) \bmod 10^9$ using the `calculate_S` function:

“`python

N = 10**11

result = calculate_S(N) % (10**9)

print(result)

“`

This will calculate the value of $S(10^{11})$ and take the modulus with $10^9$ to obtain the final result, which will be printed.

Please note that calculating the value of $S(10^{11})$ might take a long time due to the large value of $N$, so the code might require significant computational resources to execute.

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