[Guest]

Lance and his sister Alice have favorite numbers between 10 and 100, inclusive. Lance’s favorite number is divisible by Alice’s favorite number, and with digits reversed is divisible by the quantity of twenty minus Alice’s favorite number. Alice’s favorite number is prime, and is still prime with digits reversed.

Assuming that Alice's favorite number is less than twenty, find the respective favorite numbers of Lance and Alice that maximizes the absolute difference between the two's favorite numbers.

Edited January 8, 2010 by K Sengupta

[Guest]

Lance likes 99

Alice likes 11

L=99

A=11

L is divisible by A

L reversed = 99 and is divisible by 20-A = 9

A is prime and A reversed is prime

The maximum difference in the range 10 to 100 is 90

The difference between L and A is 88.

A must be prime so A cannot be 10, so if A is 11, the maximum difference in the range 11 to 100 is 89.

L must be a multiple of A so L cannot be 100. If L is 99, this is the maximum difference is 88. There can be no greater difference.