Puzzle for January 5, 2019 ( )
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, CD, and DE are 2-digit numbers (not A×B, C×D, or D×E).
Scratchpad
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Hint #1
Add C to both sides of eq.5: C + F + C = A - C + C which becomes eq.5a) 2×C + F = A Subtract 2×C from each side of the above equation: 2×C + F - 2×C = A - 2×C which becomes eq.5b) F = A - 2×C
Hint #2
In eq.2, replace A with 2×C + F (from eq.5a): 2×C + F + B + C = D + F Subtract F from each side of the above equation: 2×C + F + B + C - F = D + F - F which simplifies to eq.2a) 3×C + B = D
Hint #3
In eq.4, substitute A - 2×C for F (from eq.5b): B - C = D - E + A - 2×C Add 2×C to each side of the equation above: B - C + 2×C = D - E + A - 2×C + 2×C which becomes B + C = D - E + A Substitute 3×C + B for D (from eq.2a): B + C = 3×C + B - E + A Subtract both B and C from each side: B + C - B - C = 3×C + B - E + A - B - C which becomes 0 = 2×C - E + A Add (E - 2×C) to each side: 0 + (E - 2×C) = 2×C - E + A + (E - 2×C) which becomes eq.4a) E - 2×C = A
Hint #4
In eq.3, substitute E - 2×C for A (from eq.4a): C - D = E - 2×C - E which becomes C - D = -2×C Add 2×C + D to both sides of the equation above: C - D + 2×C + D = -2×C + 2×C + D which makes 3×C = D
Hint #5
In eq.2a, replace D with 3×C: 3×C + B = 3×C Subtract 3×C from both sides: 3×C + B - 3×C = 3×C - 3×C which makes B = 0
Hint #6
eq.6 may be written as: 10×A + B = 10×C + D + 10×D + E which becomes 10×A + B = 10×C + 11×D + E Substitute 0 for B, and (3×C) for D in the equation above: 10×A + 0 = 10×C + 11×(3×C) + E which is equivalent to 10×A = 10×C + 33×C + E which becomes eq.6a) 10×A = 43×C + E
Hint #7
In eq.6a, substitute (E - 2×C) for A (from eq.4a): 10×(E - 2×C) = 43×C + E which which is equivalent to 10×E - 20×C = 43×C + E Add (20×C - E) to each side: 10×E - 20×C + (20×C - E) = 43×C + E + (20×C - E) which simplifies to 9×E = 63×C Divide both sides by 9: 9×E ÷ 9 = 63×C ÷ 9 which means E = 7×C
Hint #8
Substitute 7×C for E in eq.4a: 7×C - 2×C = A which makes 5×C = A
Hint #9
Replace A with 5×C in eq.5b: F = 5×C - 2×C which makes F = 3×C
Solution
Substitute 5×C for A, 0 for B, 3×C for D and F, and 7×C for E in eq.1: 5×C + 0 + C + 3×C + 7×C + 3×C = 19 which simplifies to 19×C = 19 Divide both sides by 19: 19×C ÷ 19 = 19 ÷ 19 which makes C = 1 making A = 5×C = 5 × 1 = 5 D = F = 3×C = 3 × 1 = 3 E = 7×C = 7 × 1 = 7 and ABCDEF = 501373