# The wizened mathematician

Once I walked up a hill, and on the summit sat a very old man cross-legged, calmly taking the air and observing the view below. I asked him if he was alone and he said:

"No. I have five cousins who are coming up after me. What is interesting about them superficially is that if you take their ages, each one being a double-digit integer, between them they include all of the digits 0 to 9.

But if you look deeper you will see it is even more remarkable: each pair of integers within the set is not coprime, that means they have a greatest common divisor that is 2 or more."

"Very interesting indeed" I replied. "But what can I ask is the closest gap between the ages of your cousins?"

"4 years" he replied.

How old are the five cousins?

## 1 Answer

The cousins are

34, 51, 68, 72, and 90.

Pairs are divisible by

2, 3, or 17.

I hunted for

Primes with multiples consisting of 3x is and some x is or , such that f(x)+4 was divisible by 3. This seemed the most likely approach to find a solution, as it would allow a variety of combinations to post-fit test for coprimality.

• Did you also demonstrate that no other solutions are possible? (Wondering.) – Wildcard Aug 15 '17 at 1:10
• By computer I checked that the solution is unique. There are many solutions with other minimum differences: $(diff,\#sol)\in\{(3,71),(4,1),(6,92),(7,2),(9,72),(12,57),(14,42),(15,9),(16,38),(18,16),(22,1)\}$. The unique solution with a difference of 22 is $\{10,32,54,76,98\}$. – Jaap Scherphuis Aug 15 '17 at 4:26