Puzzle for December 14, 2018 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Subtract F from both sides of eq.2: C + F - F = E - F which becomes C = E - F In eq.5, replace C with E - F: A + B = E - F + D + F which becomes A + B = E + D which may be written as A + B = D + E In eq.4, replace D + E with A + B: A = A + B Subtract A from both sides: A - A = A + B - A which means 0 = B
Hint #2
In eq.5, substitute B - D + E for C + D (from eq.3): A + B = B - D + E + F Subtract B from each side: A + B - B = B - D + E + F - B which becomes A = -D + E + F In eq.4, replace A with -D + E + F: -D + E + F = D + E Add (D - E) to both sides: -D + E + F + (D - E) = D + E + (D - E) which makes F = 2×D
Hint #3
eq.6 may be written as: 10×A + B = A + 10×E + F Subtract A from each side of the equation above: 10×A + B - A = A + 10×E + F - A which becomes 9×A + B = 10×E + F Substitute 0 for B, (D + E) for A (from eq.4), and 2×D for F: 9×(D + E) + 0 = 10×E + 2×D which becomes 9×D + 9×E = 10×E + 2×D Subtract both 2×D and 9×E from each side: 9×D + 9×E - 2×D - 9×E = 10×E + 2×D - 2×D - 9×E which makes 7×D = E
Hint #4
Substitute 7×D for E in eq.4: A = D + 7×D which makes A = 8×D
Hint #5
Replace E with 7×D, and F with 2×D in eq.2: C + 2×D = 7×D Subtract 2×D from each side: C + 2×D - 2×D = 7×D - 2×D which means C = 5×D
Solution
Substitute 8×D for A, 0 for B, 5×D for C, 7×D for E, and 2×D for F in eq.1: 8×D + 0 + 5×D + D + 7×D + 2×D = 23 which simplifies to 23×D = 23 Divide each side by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = 8×D = 8 × 1 = 8 C = 5×D = 5 × 1 = 5 E = 7×D = 7 × 1 = 7 F = 2×D = 2 × 1 = 2 and ABCDEF = 805172