Consider the 6 digit substring of numbers $a*10^5+10x+b$ The first 5 consecutive digit number is $a*10^4+x$ and the second is $10x+b$ The given constraint means that 3a+x=0 mod 13 -3x+b=0 mod 13 Combining the two we get 9a+b=0 mod 13 b=4a Since 10<13 there can atmost exist one solution for b given every a. This means that the entire 30 digit number is determined by the first 5 digits. To prove that there does not exist a satisfactory 30 digit number, it suffices to prove there exists an $a$ (a does not equal 0), such that $4^{i}*a<10$ mod 13 for all $i\in{0,1,2,3,4,5}$ since there are 6 blocks of 5 digits. One can easily show that the order of 4 in mod 13 is 6. This means that $4^i*a mod 13$ for $i\in{0,1,2,3,4,5}$ either cycles through {1,4,3,12,9,10} or {2,8,6,11,5,7}. Since both of these cycles contain atleast one number that cannot be expressed as a digit (i.e. >9), there can never be a 30 digit number satisfying the given constraints.