I have to calculate modulus of $$ \frac{(7 \times 2017)!}{7^{2017}}\mod7 $$
however i have no clue how to do this. My assumption is that since both are multiplicatives of 7 that the result will be zero. However i have no way no back this up.
Thank you for taking your time.
In the numerator we have $2017$ numbers that are divisible by $7$ , but already $49$ is even divisible by $7^2$. Hence the exponent with respect to $7$ is larger than $2017$, which means that the fraction is divisible by $7$. Hence, the result is $0$. Note that your argument need not work (it would fail if the sevens would cancel out completely or the fraction would not be an integer)