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27 May 2006, 14:22
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How many numbers from 2 to 50 are not prime and are such that neither the number nor double the number is divisible by a perfect square greater than 1?
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27 May 2006, 14:33
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Solution
Five.
Since the number is not prime and is not divisible by a square greater than 1, it must be divisible by two different primes. If it were divisible by only one prime, it would either be prime itself or be divisible by the square of that prime.
Since double the number is not divisible by a square, the original number is also not divisible by 2; otherwise, its double is divisible by 4, the square of 2. Therefore, only numbers that are the product of at least two distinct primes greater than 2 satisfy the problem.
The only ones that are less than 50 are (3)(5) = 15, (3)(7) = 21, (3)(11) = 33, (3)(13) = 39, and (5)(7) = 35, so five numbers satisfy the conditions of the problem.
Five.
Since the number is not prime and is not divisible by a square greater than 1, it must be divisible by two different primes. If it were divisible by only one prime, it would either be prime itself or be divisible by the square of that prime.
Since double the number is not divisible by a square, the original number is also not divisible by 2; otherwise, its double is divisible by 4, the square of 2. Therefore, only numbers that are the product of at least two distinct primes greater than 2 satisfy the problem.
The only ones that are less than 50 are (3)(5) = 15, (3)(7) = 21, (3)(11) = 33, (3)(13) = 39, and (5)(7) = 35, so five numbers satisfy the conditions of the problem.
