Given val such that flag * 2^{10000} mod 10^{175} == val.

From definition of modulus, we get flag * 2^{10000} - k * 10^{175} == val.

Divide by 2^{175}, flag * 2^{10000-175} - k * 5^{175} == val / 2^{175}

Then we get modulo 5^{175}, flag * 2^{10000-175} == val / 2^{175} (mod 5^{175})

and multiply the inverse. flag == (val / 2^{175}) * inv(2^{10000-175}) (mod 5^{175})

Original writeup (https://gist.github.com/n-ari/33155680ecf3cf0824a29f3b77a277e6).