Product-Perfect NumbersDate: 02/26/2001 at 14:13:51 From: Mike Smith Subject: Product perfect numbers This is the question: Is there a statement to describe product perfect numbers? We call a number a product perfect number if the product of all its divisors, other than itself, is equal to the number. For example, 10 and 21 are product perfect numbers since 1*2*5 = 10 and 1*3*7 = 21, whereas 25 is not, since the product of its divisors, 1*5 = 5 is too small. I have listed all the product perfect numbers between 2-60, which are: 6,8,10,14,15,21,22,26,27,28,33,34,35,38,39,44,45,46,51,52,55,57,58. I am stumped on the description - a description that should make it easy for you to produce a six-digit product perfect number with very little work. Thank you, Mike Date: 02/26/2001 at 20:15:43 From: Doctor Schwa Subject: Re: Product-perfect numbers Hi Mike, You've made a lot of good progress. As a hint, I'll suggest that you look at the prime factorization of these numbers, and compare them to the prime factorizations of the numbers that aren't product-perfect. Perhaps by the time you get up to 20 or so you'll see a pattern... if not, please write us back for another hint. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ Date: 02/26/2001 at 23:31:29 From: Mike Smith Subject: Re: Product-perfect numbers I did all the prime factorizations of the numbers 2-20 and found that all of them have two prime factors except for 8, which is 4*2*1. I don't understand. I still can't find a pattern because 27 is 3*9 and 9 isn't prime. Can you give me another hint? Thank you, Mike Date: 02/27/2001 at 12:31:31 From: Doctor Schwa Subject: Re: Product-perfect numbers Great work! You discovered correctly that a number is perfect if, and only if, it is one of two types: either it is two different primes multiplied together, or it's a number like 8 or 27. What do those numbers have in common? Take a look at their prime factorizations... Enjoy, - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ Date: 02/27/2001 at 17:09:58 From: Mike Smith Subject: Re: Product-perfect numbers Is it that the prime factorizations are a prime and its squared number multiplied? Thank you, Mike Date: 02/27/2001 at 20:25:32 From: Doctor Schwa Subject: Re: Product-perfect numbers Exactly. A shorter way to say the same thing might be: a number is product perfect if and only if it is: either the product of two different primes, or the cube of a prime. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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