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How many 3-digit numbers are there that have exactly 16 divisors?

 Jun 22, 2022
 #1
avatar+124676 
+1

A number  written  as

 

a^15      with a being prime will have  16 divisors

But  2^15  is the smallest possible and it is   not a three  digit number

 

So.......

 

2^3 * 3 * 5 =  120 has 16 divisors                   

 

2 * 3 * 5 * 7 =  210  has 16 divisors                                

 

2^3 * 3^3  =  216  has 16 divisors                     

 

2 * 3^3   * 5  = 270 has 16 divisors                  

 

3 * 2^7  =  384    has 16 divisors                        

 

5 * 2^7 = 640  has 16 divisors                                     

 

2 * 3 * 5^3  =  750 has 16 divisors                    

 

2^7  * 7^1  =  896  has 16 divisors                     

 

{ I hope I haven't missed any  !!!   }

 

cool cool cool

 Jun 23, 2022
 #2
avatar+8 
+1

I believe you have missed some cases where all of the exponents are 1.

JustAnotherIdiot  Jun 23, 2022

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