We can list out the first few digits.
1 is the first, followed by a 9 to make the first constraint true.
Then, the 9 has to have a following 3 because 93/31 = 3. (So far is 193____)
Then after the 3 there is an 8 or 5 to make the constraint true. (So far is 1938______ or 1935______)
If 8 then impossible because no 2 digit number that starts with 8 is divisible by 31 or 19. So 1935 is accurate, after the 5 there is a 7 because 57 is divisible by 19. (So far 19357______)
After a 7 there is a 6, because 76/19 = 4. No 2 digit number with 7s as tens digit is divisible by 31 so 31 is out of this digit part. (So far is 193576___).
After the 6 there is a 2, 62/31 = 2. No 2 digit number with 6 as tens digit is div by 19. (So far is 1935762____).
After the 2, there is nothing, so unfortunately we have to start over to step 3, where instead after the 9 there is a 5. (So far is 195_____)
After the 5 is a 7. (So far is 1957_____) After the 7 there is a 6 (1957) After 7 is a 6, then after 6 is a 2. So it loops back. So whoever made this problem forgot to realize there is no such string, since no 2 digit number that starts with a 2 or 8 is divisible by either 31 or 19 :|
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