document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #684022 by KMST(5328) $\"\"$ $\"About$

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The even digits cannot be used,
\n" ); document.write( "because they would cause at least one of the 2-digit sequences to be even.
\n" ); document.write( "Similarly, the digit 5 cannot be used,
\n" ); document.write( "because it would cause at least one of the 2-digit sequences to be a multiple of 5.
\n" ); document.write( "The digits 3 and 9 cannot be used at the same time,
\n" ); document.write( "because they would cause at least one of the 2-digit sequences to be a multiple of 3 (39 or 93).
\n" ); document.write( "The two-digit sequences made with 1, 3, and 7 are all (all 6) in the list of prime numbers,
\n" ); document.write( "so $\"3%21=6\"$ 3-digit ABC sequences can be made with 1, 3, and 7.
\n" ); document.write( "Using 9, along with 1 and 7, we can also make $\"6\"$ 3-digit ABC sequences, .
\n" ); document.write( "but $\"3\"$ of the resulting ABCA sequences contain the non- prime 2-digit number $\"91=7%2A13\"$ (at the beginning, middle or end).
\n" ); document.write( "So there are $\"6%2B6-3=highlight%289%29\"$ four-digit numbers that satisfy the condition in the problem. \n" ); document.write( "

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