Is the Formula for GCD(z, nm) = d_n*d_m Correct?

[xuying1209]

For any n,m are intergers, d_n and d_m are divisors of n and m,
respectively. If gcd(n,m)=1, gcd(i,n)=d_n , gcd(j,m)=d_m
sm=1(mod n) , tn=1(mod m),
z=smi+tnj (mod nm)
then gcd(z,nm)=d_nd_m.
I want to know the result is true or false, and how to prove it.

Thanks.

 

[malaygoel]

For any n,m are intergers, d_n and d_m are divisors of n and m,
respectively. If gcd(n,m)=1, gcd(i,n)=d_n , gcd(j,m)=d_m
sm=1(mod n) , tn=1(mod m),
z=smi+tnj (mod nm)
then gcd(z,nm)=d_nd_m.
I want to know the result is true or false, and how to prove it.

Thanks.

I am not sure but the result depends on the value of s and t since they are not unique in a given problem