1991 AHSME Problems/Problem 26
An -digit positive integer is cute if its digits are an arrangement of the set and its first digits form an integer that is divisible by , for . For example, is a cute -digit integer because divides , divides , and divides . How many cute -digit integers are there?
Let the number be . We know will always divide . must divide , so must be or , but we can only use the digits to , so . must divide , so it must divide (the test for divisibility by 4 is that the last two digits form a number divisible by 4), and so must be , , , , , (not or as the is already used), or . We know divides so is even, divides so is even, and divides so is even, and thus , , and must be , , and in some order, so must be odd, so must be , , , or . Now divides implies divides , and we know divides , so must also divide , so we need a multiple of starting with or , using the remaining digits. If is , then must be , , , or , but none of these work as we need . If is , must be , , , or , so works (it has ). If is , similarly nothing works, and if is , or works. So now we have, as possibilities, , , and . But clearly can't work, as we need to be divisible by , so it must be even, so we eliminate this possibility. Now with , we need to be even, so it must be , giving , and then works as does divide . With , we get as is even, and then works as divides . Hence the number of such numbers is : and .
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