Problem:
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?
Answer Choices:
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Solution:
The fifth digit must be , and the second, fourth and sixth digits must be even. Since the first and third digits must be and in some order and but and , the second digit must be . Neither nor is divisible by , so the fourth digit must be and the sixth digit must be . Therefore there are two cute 6 -digit integers, and .