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 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 base-10 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
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Re: Minimum value of D

Quote:
 Originally Posted by CRGreathouse sod(D) = sod(D+1) + 9k - 1.
True.

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(m-1)(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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Telu Calculus 4 August 2nd, 2013 11:00 PM panky Algebra 5 November 6th, 2011 02:52 PM dk1702 Calculus 4 May 11th, 2010 06:58 PM Axle12693 Algebra 2 February 9th, 2010 04:33 PM ferret Calculus 5 October 30th, 2008 06:09 PM

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