2. The Hellenistic mathematician Nicomachus studied the following concept related to perfect numbers: A positive integer is said to be deficient if it is greater than the sum of its proper divisors and abundant if it is less than the sum of its proper divisors. Prove that there are infinitely many numbers of each type as follows: [Hint: What are its a) If p is a prime, prove that every power of p is deficient. proper divisors?] (b) If m is a positive integer, prove that 40m is abundant. [Hint: Consider first the case m = 1 and note (2) if d divides 40, then dm divides 40m, (ii) it is enough to the sum of a subset of the proper divisors of a number is greater than the number itself.] %3D

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