March 4th, 2009, 07:22 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, 08:33 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, 07:12 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.  

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