# Question: in number theory a proper divisor of a positive integer...

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In number theory, a proper divisor of a positive integer n is any x ∈ Z such that x|n and 1 ≤ x < n. A perfect number is a positive integer n that is equal to the sum of all its proper divisors. For example, 6 is perfect because 6 = 1 + 2 + 3. A deficient number is a positive integer n that is strictly greater than the sum of all its proper divisors. Fourteen is deficient because 1 + 2 + 7 = 10 < 14. An abundant number is a positive integer n that is strictly less than the sum of all its proper divisors. Thirty is abundant because 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42 > 30. (a) Is 1 perfect, deficient, abundant, or none of these? (b) What is the smallest deficient number greater than 1? (c) What is the smallest abundant number greater than 1? (d) Do you think there is a largest deficient number? Explain. (e) Do you think there is a largest abundant number? Explain. Justify your answers.