No-Solution EquationsDate: 11/09/2002 at 16:23:25 From: Mussy Subject: No-solution equations Hi, Here's the question: For any positive integer n, let S(n) denote the sum of its digits. Show that the equation n + S(n) = 1,000,000 has no solution. Then solve the equation n + S(n) = 1,000,000,000. Thank so much. Mussy Date: 11/10/2002 at 13:07:05 From: Doctor Greenie Subject: Re: No-solution equations Hi, Mussy - If we have n + S(n) = 1,000,000 then n is a 6-digit number; this means S(n) is at most 6(9) = 54. Therefore, n must be a number of the form n = 999,9AB where the digits A and B are to be determined. Rewriting our 6-digit number 999,9AB as 999,900 plus the two-digit number AB, we then have n + S(n) = 1,000,000 (999,900 + 10A + B) + (4(9) + A + B) = 1,000,000 999,936 + 11A + 2B = 1,000,000 11A + 2B = 64 This equation has no solution (A,B) with both A and B being single- digit integers. (A=4 would make B = 10; A = 5 would make B not an integer; A = 6 would make B negative.) So there is no solution to the equation n + S(n) = 1,000,000 On the other hand, if we have n + S(n) = 1,000,000,000 then n is a 9-digit number; this means S(n) is at most 9(9) = 81. Therefore, n must be a number of the form n = 999,999,9AB where the digits A and B are to be determined. Rewriting our 9-digit number 999,999,9AB as 999,999,900 plus the two-digit number AB, we then have n + S(n) = 1,000,000,000 (999,999,900 + 10A + B) + (7(9) + A + B) = 1,000,000,000 999,999,963 + 11A + 2B = 1,000,000,000 11A + 2B = 37 This equation does have a solution with A and B both single-digit positive integers: A = 3 and B = 2. So the solution for this case is n = 999,999,932 I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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