March 4th, 2009, 06:22 AM  #1 
Member Joined: Nov 2006 Posts: 54 Thanks: 0  Minimum value of D
Two consecutive positive decimal integers D and D+1 are such that the sum of the digits of each of them is divisible by 13. What is the minimum value of D? 
March 4th, 2009, 07:33 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Minimum value of D
Let sod(n) be the sum of the base10 digits of n. Unless D ends in 9 (is 9 mod 10), sod(D) + 1 = sod(D+1). If D ends in 9, but not in 99, sod(D) = sod(D+1) + 8. Generalizing, if D ends in precisely k (not k+1) 9s, sod(D) = sod(D+1) + 9k  1. Can you solve it from here? 
March 5th, 2009, 06:12 AM  #3  
Member Joined: Nov 2006 Posts: 54 Thanks: 0  Re: Minimum value of D Quote:
From this point onwards, we note that since each of sod(D) and sod(D+1) is divisible by 13, it follows that 9k – 1 is divisible by 13. The minimum value of k for which this is possible occurs at k=3. Accordingly, D = X1X2….Xm999, where none of X1, X2, …., Xm is 9. > D+1 = X1X2….X(m1)(Xm + 1)000, with the restriction that: (1 + Sum(i=1 to m) Xi) is divisible by 13. We now observe that (D+1) minimized whenever m=2, with: (X1, X2) = (4,. Therefore, D+1 = 49000, giving: D = 48999 Consequently, the minimum value of D in conformity with the given conditions is 48999.  

Tags 
minimum 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Help with a minimum  Telu  Calculus  4  August 2nd, 2013 10:00 PM 
minimum value  panky  Algebra  5  November 6th, 2011 01:52 PM 
Minimum value  dk1702  Calculus  4  May 11th, 2010 05:58 PM 
Minimum value?  Axle12693  Algebra  2  February 9th, 2010 03:33 PM 
Minimum value  ferret  Calculus  5  October 30th, 2008 05:09 PM 