If a(n) =kn, where k ::=::: 3, then the positive integer n is called a k-perfect

Chapter 11, Problem 9

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If a(n) =kn, where k ::=::: 3, then the positive integer n is called a k-perfect number(sometimes, multiply perfect). Establish the following assertions concerning k-perfectnumbers:(a) 523776 = 29 3 11 31 is 3-perfect.30240 = 25 33 5 7 is 4-perfect.14182439040 = 27 34 5 7 112 17 19 is 5-perfect.(b) If n is a 3-perfect number and 3 l n, then 3n is 4-perfect.(c) If n is a 5-perfect number and 5 l n, then 5n is 6-perfect.(d) If 3n is a 4k-perfect number and 3 J n, then n is 3k-perfect.For each k, it is conjectured that there are only finitely many k-perfect numbers. Thelargest one discovered has 558 digits and is 9-perfect.

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