Puzzle for April 9, 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, BC, and EF are 2-digit numbers (not A×B, B×C, or E×F).
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Hint #1
In eq.2, replace B with D - E (from eq.5): D - E + C = D + E Add E to both sides of the above equation: D - E + C + E = D + E + E which becomes D + C = D + 2×E Subtract D from both sides: D + C - D = D + 2×E - D which makes C = 2×E
Hint #2
In eq.3, replace C with 2×E: A = 2×E + E which makes A = 3×E
Hint #3
Substitute 2×E for C, and D - E for B (from eq.5) in eq.4: 2×E + D = D - E + F Add E to both sides, and subtract D from both sides of the above equation: 2×E + D + E - D = D - E + F + E - D which simplifies to 3×E = F
Hint #4
eq.6 may be written as: 10×A + B = 10×B + C + 10×E + F Subtract B from each side of the above equation: 10×A + B - B = 10×B + C + 10×E + F - B which becomes eq.6a) 10×A = 9×B + C + 10×E + F
Hint #5
Substitute 3×E for both A and F, and 2×E for C in eq.6a: 10×3×E = 9×B + 2×E + 10×E + 3×E which becomes 30×E = 9×B + 15×E Subtract 15×E from each side: 30×E - 15×E = 9×B + 15×E - 15×E which makes 15×E = 9×B Divide both sides by 9: 15×E ÷ 9 = 9×B ÷ 9 which makes 1⅔×E = B
Hint #6
Substitute 1⅔×E for B in eq.5: D - E = 1⅔×E Add E to both sides: D - E + E = 1⅔×E + E which makes D = 2⅔×E
Solution
Substitute 3×E for A and F, 1⅔×E for B, 2×E for C, and 2⅔×E for D in eq.1: 3×E + 1⅔×E + 2×E + 2⅔×E + E + 3×E = 40 which simplifies to 13⅓×E = 40 Divide both sides by 13⅓: 13⅓×E ÷ 13⅓ = 40 ÷ 13⅓ which means E = 3 making A = F = 3×E = 3 × 3 = 9 B = 1⅔×E = 1⅔ × 3 = 5 C = 2×E = 2 × 3 = 6 D = 2⅔×E = 2⅔ × 3 = 8 and ABCDEF = 956839