20200320, 14:35  #1 
Mar 2018
17·31 Posts 
216 is the only cube that can be written in this way?
216=3^2*(182^2331*10^2)
is there any other positive cube that can be written as a^2*(b^2r*10^2)? with a and b positive and coprime, r prime? 
20200320, 15:16  #2  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2×3×5×353 Posts 
Quote:
It should now be obvious that there are an infinite number of solutions. Picking just one pretty much at random, let r =1. In that case b^2 > 100, or b> 10. One such value is 11. Plugging in the numbers, d=121100 = 21. Indeed 21^3 = 9261 = 441 * (11^2  10 * 1^2). Why do you post questions which are so trivial to answer with even the slightest amount of thought? Last fiddled with by xilman on 20200320 at 15:20 

20200322, 11:13  #3 
Romulan Interpreter
Jun 2011
Thailand
22073_{8} Posts 

20200322, 15:30  #4  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2·3·5·353 Posts 
Quote:
OK, choose r=2. Prime, right? Let b=17, so B^2100r = 289200 = 89 = a. All are positive and coprime. Let b=19, so b^2100r = 361200 = 161 = 7 * 23 = a. All are positive and coprime. Let b=21, so a=241 which is a prime. And so on. It is still a trivial problem. 

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