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 and solutions on this page are the property of the MAA's American Mathematics Competitions