### Brain Teasers

# Prime Ponderings

What are the largest and smallest 5-digit numbers that satisfy the following conditions?

1. Each digit of the number is a prime digit.

2. Each successive pair of digits forms a 2-digit number that is NOT a prime number.

3. Each of the prime digits must appear at least once in the 5-digit number.

1. Each digit of the number is a prime digit.

2. Each successive pair of digits forms a 2-digit number that is NOT a prime number.

3. Each of the prime digits must appear at least once in the 5-digit number.

### Answer

Largest - 35772Smallest - 32257

Prime digits are 2, 3, 5 and 7. Taken as pairs, the only combinations that fit criterion 2 are:

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75 and 77.

Of that list, 33 is the only number that contains a 3 as the second digit and so for criterion 3 to be satisfied, 3 can only appear at the start of the 5-digit numbers. From there it is simply a matter of choosing the next digits using the list above so that criterion 2 is satisfied, making sure that criterion 3 is also satisfied.

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