CRA[imgs_List,mods_List]:=Module[{L={{imgs[[1]],mods[[1]]}},x=imgs[[1]],m=1,s,k},
Do[m=m mods[[k-1]];s=PolynomialMod[1/m,mods[[k]]];
x=x+Mod[(imgs[[k]]-x)s,mods[[k]]] m;AppendTo[L,{x,m mods[[k]]}]
,{k,2,Length[imgs]}];Return[L];];

goal=RandomInteger[10^35]

primes=NextPrime[2^31,-Range[7]]

imgs=Mod[goal,primes]

TableForm[Prepend[CRA[imgs,primes],{"x","m" Subscript["p","k"]}]]

Newton[imgs_,mods_,x_]:=Module[{L={{imgs[[1]],x-mods[[1]]}},p=imgs[[1]],m=1,s,k},
Do[m=m (x-mods[[k-1]]);s=1/m/.x->mods[[k]];
p=p+((imgs[[k]]-p)s/.x->mods[[k]]) m;AppendTo[L,{p,m (x-mods[[k]])}]
,{k,2,Length[imgs]}];Return[L];];

goal=Sum[RandomInteger[10]x^i,{i,0,5}]

imgs=goal/.({x->#}&/@Range[10])

TableForm[Prepend[MapAt[Expand,#,1]&/@Newton[imgs,Range[10],x],{"p","m"(x-k)}]]

TableForm[Prepend[MapAt[Factor,#,1]&/@Newton[{0,0,0,1,0,0},Array[u,6],x],{"p","m"(x-u[k])}]]