Puzzle for April 12, 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 E to both sides of eq.3: B + C + D - E + E = A + F + E which becomes B + C + D = A + F + E which may be written as B + C + D = A + E + F In the above equation, replace E + F with A + D (from eq.2): B + C + D = A + A + D Subtract D from each side of the above equation: B + C + D - D = A + A + D - D which becomes eq.3a) B + C = 2×A
Hint #2
In eq.3a, replace C with A + B (from eq.4): B + A + B = 2×A Subtract A from both sides of the above equation: B + A + B - A = 2×A - A which means 2×B = A
Hint #3
In eq.4, substitute 2×B for A: C = 2×B + B which makes C = 3×B
Hint #4
Substitute 2×B for A, and 3×B for C in eq.5: F - 3×B = 2×B + 3×B + E Add 3×B to both sides of the equation above: F - 3×B + 3×B = 2×B + 3×B + E + 3×B which becomes eq.5a) F = 8×B + E
Hint #5
Substitute 2×B for A, and 8×B + E for F (from eq.5a) in eq.2: 2×B + D = E + 8×B + E Subtract 2×B from each side: 2×B + D - 2×B = E + 8×B + E - 2×B which becomes eq.2a) D = 6×B + 2×E
Hint #6
eq.6 may be written as: 10×D + E - (10×C + D) = 10×A + B + C which may be written as 10×D + E - 10×C - D = 10×A + B + C Subtract C from both sides of the above equation: 10×D + E - 10×C - D - C = 10×A + B + C - C which becomes eq.6a) 9×D + E - 11×C = 10×A + B
Hint #7
Substitute (6×B + 2×E) for D (from eq.2a), 3×B for C, and 2×B for A in eq.6a: 9×(6×B + 2×E) + E - 11×3×B = 10×2×B + B which may be written as 54×B + 18×E + E - 33×B = 20×B + B which becomes 21×B + 19×E = 21×B Subtract 21×B from each side: 21×B + 19×E - 21×B = 21×B - 21×B which becomes 19×E = 0 which means E = 0
Hint #8
Substitute 0 for E in eq.2a: D = 6×B + 2×0 which makes D = 6×B
Hint #9
In eq.5a, replace E with 0: F = 8×B + 0 which means F = 8×B
Solution
Substitute 2×B for A, 3×B for C, 6×B for D, 0 for E, and 8×B for F in eq.1: 2×B + B + 3×B + 6×B + 0 + 8×B = 20 which simplifies to 20×B = 20 Divide both sides by 20: 20×B ÷ 20 = 20 ÷ 20 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 3×B = 3 × 1 = 3 D = 6×B = 6 × 1 = 6 F = 8×B = 8 × 1 = 8 and ABCDEF = 213608