Solution: RSA modulus

n: 275576263849684251633432947333044480211
e: 17
ciphertext: 79395419347137974212436862718863190330

flag is rsa_<decrypted number>

The given public modulus 275576263849684251633432947333044480211 is small enough that it's trivial for a modern computer to factor. Factored, it yields:

p: 14974393262425108487
q: 18403167261619959253

Use a tool (or just some plain ol' math) to calculate the private key d from p, q, and e.

d: 105367395001349860905903530250499775357
      (with Carmichael's totient)
or 243155526926191986705931223654999481593
      (with Euler's totient)

Decrypting the ciphertext with d and n, we find the plaintext to be 83295603.

The flag:

rsa_83295603

For an explanation of how to actually do the operations discussed above, see RSA (cryptosystem) on Wikipedia.

Online tools for factoring:

• Alpertron's integer factorization calculator

Online tools for calculating:

• My own RSA calculator