2005 PMWC Problems/Problem T10

Problem

Find the largest 12-digit number for which every two consecutive digits form a distinct 2-digit prime number.

Solutions

First Solution:

We list all 2 digit primes:

11, 13, 17, 19

23, 29

31, 37

41, 43, 47

53, 59

61, 67

71, 73, 79

83, 89

97

Picking a 9 would get us:

97371311

So we pick an 8.

837319737131

That's the greatest.

[Comment: In this solution the number 837319737131 has many non-distinct two digit subsequences of prime numbers for example 37, 31, 73 all appear more than once. But the question asks for distinct two digit prime numbers.]

Second Solution:

We start with 97, which is the largest 2 digit prime.

97

Then we add 9 to get 79, the largest 2 digit prime with tens digit 7.

979

Add 3 to get 93, the largest prime less than 97

9793

Now the largest two digit prime with tens digit 3 is 37. So we add a 7

97937

Now we add another 3 as 79 already exists.

979373

And proceeding like this will get us

979373191713

Which answer is correct?