Question 1068756: Let A, B, and C represent distinct digits. A four-digit positive integer of the form ABCA has the property that the two-digit integers AB, BC, and CA are all primes. Compute the number of all such four-digit integers ABCA.
Found 2 solutions by Edwin McCravy, KMST:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
All the 2 digit prime numbers are:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97
All 4-digit numbers of the form ABCA, where A,B, and C are
all different and AB, BC, and CA are all prime numbers are:
1371, 1731, 1971, 3173, 3713, 7137, 7197, 7317, 9719,
Edwin
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! The even digits cannot be used,
because they would cause at least one of the 2-digit sequences to be even.
Similarly, the digit 5 cannot be used,
because it would cause at least one of the 2-digit sequences to be a multiple of 5.
The digits 3 and 9 cannot be used at the same time,
because they would cause at least one of the 2-digit sequences to be a multiple of 3 (39 or 93).
The two-digit sequences made with 1, 3, and 7 are all (all 6) in the list of prime numbers,
so  3-digit ABC sequences can be made with 1, 3, and 7.
Using 9, along with 1 and 7, we can also make  3-digit ABC sequences, .
but  of the resulting ABCA sequences contain the non- prime 2-digit number  (at the beginning, middle or end).
So there are  four-digit numbers that satisfy the condition in the problem.
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