Puzzle for December 15, 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.
* CD is a 2-digit number (not C×D).
Scratchpad
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Hint #1
Subtract F from each side of eq.4: D + F - F = E - F which becomes eq.4a) D = E - F Add C to both sides of eq.6: F - C + C = B + C + C which becomes eq.6a) F = B + 2×C
Hint #2
Add F to both sides of eq.2: A + B + F = C + D + E + F Replace B + F with CD (from eq.5) in the above equation: A + CD = C + D + E + F which may be written as A + 10×C + D = C + D + E + F Subtract both C and D from each side: A + 10×C + D - C - D = C + D + E + F - C - D which becomes eq.2a) A + 9×C = E + F
Hint #3
In eq.2a, replace F with B + 2×C (from eq.6a): A + 9×C = E + B + 2×C Subtract 2×C from each side: A + 9×C - 2×C = E + B + 2×C - 2×C which becomes A + 7×C = E + B Substitute A + C for B (from eq.3): A + 7×C = E + A + C Subtract both A and C from each side: A + 7×C - A - C = E + A + C - A - C which makes 6×C = E
Hint #4
Replace D with E - F (from eq.4a) in eq.2: A + B = C + E - F + E which becomes A + B = C + 2×E - F Substitute (6×C) for E, and (B + 2×C) for F (from eq.6a) in the above equation: A + B = C + 2×(6×C) - (B + 2×C) which becomes A + B = C + 12×C - B - 2×C which becomes A + B = 11×C - B Add B to each side: A + B + B = 11×C - B + B which becomes eq.2b) A + 2×B = 11×C
Hint #5
From eq.3, substitute (A + C) for B in eq.2b: A + 2×(A + C) = 11×C which is the same as A + 2×A + 2×C = 11×C Subtract 2×C from each side of the above equation: A + 2×A + 2×C - 2×C = 11×C - 2×C which simplifies to 3×A = 9×C Divide both sides by 3: 3×A ÷ 3 = 9×C ÷ 3 which makes A = 3×C
Hint #6
Replace A with 3×C in eq.3: 3×C + C = B which makes 4×C = B
Hint #7
In eq.6a, replace B with 4×C: F = 4×C + 2×C which makes F = 6×C
Hint #8
In eq.4a, replace both E and F with 6×C: D = 6×C - 6×C = 0
Solution
Substitute 3×C for A, 4×C for B, 0 for D, and 6×C for E and F in eq.1: 3×C + 4×C + C + 0 + 6×C + 6×C = 20 which simplifies to 20×C = 20 Divide both sides by 20: 20×C ÷ 20 = 20 ÷ 20 which means C = 1 making A = 3×C = 3 × 1 = 3 B = 4×C = 4 × 1 = 4 E = F = 6×C = 6 × 1 = 6 and ABCDEF = 341066