Problem:
Let be the smallest positive integer such that is divisible by is a perfect cube, and is a perfect square. What is the number of digits of ?
Answer Choices:
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Solution:
Because is divisible by , where and are nonnegative integers and is a positive integer not divisible by 2 or 5 . Because is a perfect cube, 3 divides and 3 divides . Because is a perfect square, 2 divides and 2 divides . Therefore 6 divides and 6 divides . The smallest possible choices for , and , are , and . In this case , and has digits.
OR
The only prime factors of 20 are 2 and 5 , so has the form for integers and . Because is a perfect cube, and are both multiples of 3 , so and are also both multiples of 3 . Similarly, because is a perfect square, and are both multiples of 2 . Therefore both and are multiples of 6. Note that satisfies the given conditions, and has digits.
The problems on this page are the property of the MAA's American Mathematics Competitions