Find all positive integers that are precisely 2007 times their respective sum of the digits.
*** For an extra challenge, solve this puzzle without using a computer program aided method.
If K= Sum of digits of N
N=K*2007 => N = 0 (mod 9) => K= 0 (mod 9) (necessary  not sufficient)
We try:
K=9 N=K*2007 = 18063 => K=18 (contradictory)
K=18 N=K*2007 = 36126 => K=18 (valid value)
K=27 N=K*2007 = 54189 => K=27 (valid value)
K=36 N=K*2007 = 72252 => K=18 (contradictory)
From here higher values of K (45, 54, 63...) lead to impossibility as a 5 digit N which is multiple of 2007 has his digit sum less than 45; a 6 digit less than 54, so that other matches will be impossible.

Posted by armando
on 20160527 09:20:37 