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A Perfect Surprise

Consider this rule: N is an integer such that the sum of the reciprocals of its divisors equals 2.

Your Task: Find the three smallest integers N that satisfy this rule.


Source: Adapted from Mathematics Magazine, 1954, pp. 37-38.

Hint: Can the integer N be prime?

Try some numbers with multiple factors...being systematic may or may not help.


Solution Commentary: For N (and its divisors dn), we need Σ1/dn = 2. Suppose we multiplied the left expression by N, getting Σ N/dn = Σ dn. Think about why this works…try some examples. But this means that Σ dn = 2N. Surprise….this means N is a perfect number (i.e. a number that is the sum of its proper divisors). Thus, the three smallest perfect numbers are 6, 28, and 496.