There are 362,880 different ways to arrange the digits 1 through 9 into a 9-digit number. How many of these combinations are prime numbers?


None of the combinations are prime numbers.

There are a couple of ways to show this, but the easiest is using the divisibility rule for 3. We know that if the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3 (and vice versa).

All 9-digit combination of the digits 1 through 9 have the same sum of digits 1+2+3+4+5+6+7+8+9=45, which is divisible by 3. Therefore all of these combinations are divisible by 3 and none of them are prime numbers.

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