Self Powers

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:

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

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