The digital root of a number is obtained by summing its digits and then repeating the process (summing the sum's digits) until the answer is a single digit number. For example, the digital root of 7368 is 6.

Determine the digital root of (9⁷²⁴⁰+38)⁸⁴⁵⁷

from the two properties given in my hint

let p=7240 and q=8457

s( (38+9^p)^q ) = s( s(38+9^p)^q )

s(38+9^p) = s( s(38) + s(9^p) )

digital root of a multiple of 9 is always 9 thus

s(38+9^p) = s( 2+9 ) =s(11)=2

thus
s( (38+9^p)^q ) = s( 2^q)

2^q=11267

1+1+2+6+7=17
1+7=8

thus s(2^q) = 8
thus s((38+9^p)^q)=8

and thus our answer is 8

sorry I'm in a bit of rush to get to breakfast with my wife so I'll be posting my proof for the two properties used at a later time :-D

Posted by Daniel on 2006-06-29 11:48:24