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Math Help - Divisor of perfect number is not perfect

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    Divisor of perfect number is not perfect

    Show that no proper divisor of a perfect number is perfect.

    proof. let n be perfect, and let d_{1},d_{2},...,d_{r} be divisors of n. Now, O(n) = 2n = 1 + d_{1},d_{2},...,d_{r} .

    Now, what should I do to get O(d) not equal to 2d for all ds?
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  2. #2
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    Try proving the contrapositive: if d is perfect and n=kd is a multiple of d, then n is not perfect. To see this, notice that if d is the sum of its proper divisors, say d=d_1+\ldots+d_r, then n=kd_1+\ldots+kd_r. So n is the sum of some of its proper divisors. But n also has other divisors that are not multiples of k (for example, 1). Therefore the sum of the proper divisors of n is greater than n.
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