## Related questions with answers

Catrina selects three integers from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and then forms the six possible three-digit integers (leading zero allowed) they determine. For instance, for the selection 1, 3, and 7, she would form the integers 137, 173, 317, 371,713, and 731. Prove that no matter which three integers she initially selects, it is not possible for all six of the resulting three-digit integers to be prime.

Solution

VerifiedSince an even number on units digit makes the resultant number even ($> 2$), thus composite, we can safely ignore the even numbers from the set. Also any number at least 10 and having 5 at its units digit must be divisible by 5 even though the number itself is greater than 5, we can also ignore 5. Thus the remaining of the set is $\{ 1, 3, 7, 9\}$.

Since $7 | n \Leftrightarrow 7 | \frac{n-u}{10} -2u$, where $u$ is the units digit of $n$, the result follows from the fact that $7 | 37 - 2.1, 7 | 93 - 2.1, 7 | 79 - 2.1, 7 | 97 - 2.3$

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