The number of divisors of the form  $4n+1,n\in N$ of the number  ${10}^{10}\cdot {11}^{11}\cdot {13}^{13}$ contains the digit(s):

Any factor of  ${10}^{10}\cdot {11}^{11}\cdot {13}^{13}=2^{10}\cdot 5^{10}\cdot {11}^{11}\cdot {13}^{13}$  will be of the form  $2^a{5}^b{11}^c{13}^d$ where  $a=0,1,2,\mathrm{...10};b=0,1,2,\mathrm{...10};c=0,1,2,\mathrm{...11};d=0,1,2,\mathrm{...13};$ 
Factors should be of the type $4n+1\Rightarrow a=0;c=0,2,4,6,8,10;$   and  $b,d$ can be any possible value.
Number of such divisors  $=1\cdot 6\cdot 11\cdot 14=924$
Number of divisors of the form  $4n+1,n\in N$ will be 923