.
Ursula has the tiles numbered as 1, 3, 2, 6. How many different ways can she arrange the tiles to form numbers that are divisible by 6?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The necessary and sufficient condition for a number to be divisible by 6
is to be even and to be divisible by 3.
So, the last digit must be 2 or 6. Then and only then the number is even.
From the other side, the sum of the digits 1 + 3 + 2 + 6 = 12 is divisible by 3. Hence, any permutation of these digits is the number divisible by 3.
Thus any permutation of the digits that ends by 6 or by 2 is multiple of 6.
All other permutations are not multiples of 6.
Next, there are 3! permutations ending by 2, and
there are 3! permutations ending by 6.
In all, there are 3! + 3! = 12 numbers of these digits that are multiple of 6.
Answer. There are 12 numbers of these digits that are multiple of 6.