Is it possible to form six prime numbers using three distinct digits?

  • MHB
  • Thread starter Ackbach
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    2015
In summary, "forming six prime numbers using three distinct digits" means finding six different combinations of three digits that, when arranged in different orders, create six prime numbers. It is not possible to form six prime numbers using any three digits, as there are certain combinations that will not result in any prime numbers. A prime number is a number that is only divisible by 1 and itself, and there are some patterns and rules that can help in forming prime numbers using three distinct digits. For example, if the sum of the three digits is a multiple of 3, or if the last digit is even, then the resulting numbers will not be prime. One example of three digits that can form six prime numbers is 1, 4,
  • #1
Ackbach
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Here is this week's POTW, which marks my first anniversary of being the University POTW Director:

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Given three distinct digits, six numbers can be formed such that each of the given digits appears exactly once in any of them; e.g., using 1, 2, and 5, you can form 125, 152, 215, 251, 512, and 521. Is it possible to choose the three digits in such a way that all of the six numbers so formed are prime?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
We had excellent participation this week, with kiwi, johng, and petek all providing correct solutions. Congratulations to you!

petek's solution follows below:

None of the digits can equal 0, 2, 4, 6, 8 or 5, since a permutation ending in one of those digits will be divisible either by 2 or 5. Therefore, the three digits must occur in the set {1, 3, 7, 9}. The four possibilities of three digits are {1, 3, 7}, {1, 3, 9}, {1, 7, 9} or {3, 7, 9}. However, each of these sets of three digits contains at least one permutation that's composite:

371 = 7 x 53
931 = 7^2 x 19
791 = 7 x 113
973 = 7 x 139

(Some other permutations also are composite.)
 

Related to Is it possible to form six prime numbers using three distinct digits?

1. Can you explain what "forming six prime numbers using three distinct digits" means?

This means finding six different combinations of three digits that, when arranged in different orders, create six prime numbers. For example, using the digits 1, 2, and 3, the numbers 123, 132, 213, 231, 312, and 321 are all prime numbers.

2. Is it possible to form six prime numbers using any three digits?

No, it is not possible to form six prime numbers using any three digits. There are certain combinations of three digits that will not result in any prime numbers.

3. How do you determine if a number is prime or not?

A prime number is a number that is only divisible by 1 and itself. To determine if a number is prime, you would need to check if it is divisible by any number other than 1 and itself. If it is not divisible by any other number, then it is a prime number.

4. Are there any patterns or rules for forming prime numbers using three distinct digits?

Yes, there are some patterns and rules that can help in forming prime numbers using three distinct digits. For example, if the sum of the three digits is a multiple of 3, then none of the resulting numbers will be prime. Also, if the last digit is even, then the resulting number will not be prime.

5. Can you give an example of three digits that can form six prime numbers?

One example is the digits 1, 4, and 7. When arranged in different orders, they can form the prime numbers 147, 174, 417, 471, 714, and 741.

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