Prove that no number in the sequence 371, 3711, 37111... is prime.

t is easy to show that 37 followed by n "ones" is always a composite, since the resulting number is divisible by at least one of the following primes: 3, 7, 13, 37:

If n= 0 mod3 then it is divisible by 37 since 111=3*37

If n= 2 mod3 then it is divisible by 3 since the sod is a multiple of 3

If n= 1 mod6 then it is divisible by 7 since 111111=3*37*7*11*13 and 371=7*53

If n= 4 mod6 then it is divisible by 13 since111111=3*37*7*11*13

and 371111=260000+111111

Bonus finding: the same applies to 38 in lieu of 37: 381, 3811, 38111…Prove it.