## The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$.

Find the last ten digits of the series, $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$.

### The key idea is to use modular arithmetic to work with the last ten digits of each term individually, and then accumulate the sum while keeping the result modulo $10^{10}$ at each step.

Here’s how you can approach this problem using Python:

,,,,,,,,,,

def modular_pow(base, exponent, modulus):

result = 1

base %= modulus

while exponent > 0:

if exponent % 2 == 1:

result = (result * base) % modulus

exponent >>= 1

base = (base * base) % modulus

return result

def main():

modulus = 10 ** 10

total_sum = 0

for n in range(1, 1001):

total_sum = (total_sum + modular_pow(n, n, modulus)) % modulus

print(total_sum)

if __name__ == “__main__”:

main()

,,,,,,,,

Running this code will calculate and print the last ten digits of the series $1^1 + 2^2 + 3^3 + \cdots + 1000^{1000}$.

##### More Answers:

Triangular, Pentagonal, and HexagonalGoldbach’s Other Conjecture

Distinct Primes Factors

Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded