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Find all integers \(n\) such that the arithmetic mean of all its factors is an integer.

 

Thanks so much!

 Mar 13, 2022
 #1
avatar+23200 
+1

Every odd prime number has two factors and only two factors, 1 and the prime number.

The result of adding 1 to the odd prime number is an even number; dividing this 

answer by 2 results in an integer.

Now, you have an infinite number of answers.

And yet, there are more; some even numbers also fulfill the requirement ...

 Mar 14, 2022
 #2
avatar+31 
+1

Oh! That's really cool! I haven't thought of the odd primes.

 

I also assume that some non-odd primes work and as you say, some even numbers work too. Is there a specific way of describing them or is it just at random?

 

Thank you so much,geno3141!

ibhar  Mar 14, 2022
 #3
avatar+117852 
+1

Thanks Geno,

I think your answer is cool too :)

 

Attn:  ibhar,

Thank you for responsing to Gino's answer so nicely.   laugh

BUT

You need to think.

Your statement:   "I also assume that some non-odd primes work"

1) Firstly, tell me some non-odd primes.

2) second show me with one of them what the average of its factors is. 

3) discuss your answers.

 

YES  you are expected to answer me :)

Melody  Mar 14, 2022
edited by Melody  Mar 14, 2022

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