Two Digit Primes

The digits 1, 2, 3, 4, 6, 7, 8, and 9 are used to form four two-digit prime numbers.  If each digit is used only once, find the sum of the four two-digit prime numbers.
Source: mathcontest.olemiss.edu 11/17/2008

SOLUTION
The two-digit prime numbers are: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 83, 89, and 97.

Since each of the even digits {2, 4, 6, 8} is used only once, we form the four numbers as follows:

2\;\_\_ with {3, 9} to fill in the unit digit

4\;\_\_ with {1, 3, 7} to fill in the unit digit

6\;\_\_ with {1, 7} to fill in the unit digit

8\;\_\_ with {3, 9} to fill in the unit digit

Choose 23
If we choose 23, then the next number could be 41 or 47. We now have two paths:

Path 1: 23, 41, 67, 89 which sum to 23+41+67+89=220.

Path 2: 23, 47, 61, 89 which sum to 23+47+61+89=220.

Choose 29
If we choose 29, then the next number could be 41, 43, or 47. We now have three paths:

Path 1: 29, 41, 67, 83 which sum to 29+41+67+83=220.

Path 2.1: 29, 43, 61, 8 __ . There is no solution.
Path 2.2: 29, 43, 67, 8 __. There is no solution.

Path 3: 29, 47, 61, 83 which sum to 29+47+61+83=220.

Answer: 220.

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About mvtrinh

Retired high school math teacher.
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One Response to Two Digit Primes

  1. Chris West says:

    Nice post!

    For all those that are improving their math skills, I think that it is important to note that since you could only use 1,3,7, and 9 in the ones-digit and 2,4,6, and 8 in the tens-digit, no matter how you formed the prime numbers, the sum would be the same. Of course, this would be the case even if you just stated that the numbers had to be odd (instead of prime).

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