I think such an exploration can be a great independent exploration for interested and capable students, and so I will expand a little on the question.
It is obvious, I think that extending 11 by adding more ones will quickly produce a multiple of three. And 1111 and 11111 as well as 1111111 are all composite. But extending 13 to 133, 1333, and 1333 are not prime (although they are all semi-primes (two prime factors) and students might wonder how far, or how often that is true for more 3's) . 17 undergoes a similar failure to produce primes, although 1777 is prime. But 19, 199, and 1999 are all prime. Unfortunately, 19999 is 7 x 2857.
With 23, 233, 2333, 23333 are again all prime. Is it possible for this method to produce more than four primes in a row. We might look instead at extending the first digit, 113 1113, etc for 13, and similarly for the first digits of other two digit numbers. Conversly, they might explore how many times the digit might be extended before another primes is found... will 1333..3 EVER be prime?... and will there always be another prime if more digits are added?
For more than two digits there are extended options. We could repeat the last digit, extending xyz to xyzz, or alternately replicating the last two digits, xyzyz.
For students working on divisibility schemes, testing primes, etc this is probably great practice, and for others, it cold just be good fun.
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