Checking the Divisibility of a permutation

Probability Level 3

Let $S$ denote the set of numbers where each number is a permutation of the digits $1,2,3,4,5,6,7,8$ . A number $x$ is chosen randomly from the set $S$ . The probability that $x$ is divisible by 36 is $\frac mn$ where $m$ and $n$ are positive coprime integers. Find the value of $m+n$ .

The answer is 5.

1 solution

Hillary Diane Andales
May 18, 2014

To be divisible by 36, the numbers must be divisible by both 9 and 4. All the permutations of the digits are divisible by 9. Only one more condition needs to be satisfied: divisibility by 4.

Permutations divisible by 4 must end in 12, 16, 24, 28, 32, 36, 48, 52, 56, 64, 68, 72, 76, and 84. For each of 12, 16, 24, etc, there are 6! permutations. The probability now is:

$\frac{6! (14)}{8!} = \frac{14}{56} = \frac{1}{4}$

So, $m + n = \boxed{5}$

Perfect. :D

Finn Hulse - 7 years ago

Thanks @Finn Hulse :D I really admire your solutions by the way.

Hillary Diane Andales - 7 years ago

HAHAHA compliment accepted but they're not that great. But thanks anyways! :D

Finn Hulse - 7 years ago

Checking the Divisibility of a permutation

The answer is 5.

1 solution

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