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Believe it or not 2 (Posted on 2021-12-30) Difficulty: 3 of 5
There are only 4 positive integers N such that
multiplying sod(N) by inverted(sod(N)) equals N.
The trivial number 1 qualifies !
The magic Ramanujan number 1729 qualifies as well ,
Since 19*91=1729

Now, find the missing two and prove that no other exist.

See The Solution Submitted by Ady TZIDON    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 1 of 2
for n=1:5000000
   digs=char(string(n));
   s1=sod(n);
   s2=flip(char(string(s1)));
   if s1*str2double(s2)==n
       disp(n)
   end
end

function sd = sod(n) 
    s=0;
    dgts=num2str(n) ;
    l=length(dgts) ;
    for i=1:l
        s=s+str2double(dgts(i));
    end
    sd=s;
end

finds only

     1
    81
  1458
  1729
  
and in fact the checking to 5 million is overkill:

Maximum SODs are proportional to log(n), and the product of two SODs to the square of log(n). At five digit n, the maximum SOD is 9*5=45, whose square is 2025, and thus already too small to match a 5-digit number. This is all the more so for higher numbers of digits; for example (9*6)^2 = 2916, too small to match a 6-digit number.

  Posted by Charlie on 2021-12-30 09:08:52
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