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   Author  Topic: A Digital Puzzle  (Read 1343 times)
K Sengupta
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A Digital Puzzle  
« on: Mar 1st, 2007, 5:55am »		 Quote Quote Modify Modify
Analytically determine:
 
 (i) the digital root of S and,  
 
(ii) the  multiplicative digital root of S;
 
where:
 
S = [(22007 + 35)/225].
 
[x] denotes the greatest integer <=x
 
NOTE:  
 
The definition of digital root is given in http://mathworld.wolfram.com/DigitalRoot.html while the definition of multiplicative digital root is given in : http://mathworld.wolfram.com/MultiplicativeDigitalRoot.html
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Eigenray
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Re: A Digital Puzzle  
« Reply #1 on: Mar 8th, 2007, 4:37pm »		 Quote Quote Modify Modify
(i)  Since phi(34)=34*(1-1/3) = 54, 254 = 1 mod 34, and since 2007 = 9 mod 54, 22007 = 29 = 512 = 26 mod 34.  In particular, 22007 = 8 mod 9.
 
Similarly, phi(52) = 20, so 22007 = 27 = 128 = 3 mod 52.
 
So 22007 is 8 mod 9 and 3 mod 25.  Using the Chinese remainder theorem, or by inspection, 22007 = 53 mod 225, and so
 
S = [(22007+35)/225] = (22007 - 53)/225.
 
Now, 22007 - 53 = 26 - 53 = -27 mod 34, so
 
S = -27/225 = -3/25 = 3/2 = 6 mod 32,
 
and the digital root is 6.
 
(ii)  This one I'm not sure how to do by hand, but it suffices to compute
 
log10 S ~ log10 (22007/225) = 2007log10 2 - log10 225 = 601.815,
 
so S ~ 10601.815 ~ 6.5 * 10601, so the product of the digits of S is 6*5*..., hence ends in 0, so the multiplicative digital root of S is 0.
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