Puzzle for June 14, 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 D and E to both sides of eq.5: F - D + D + E = F - E + D + D + E which becomes F + E = F + 2×D Subtract F from each side: F + E - F = F + 2×D - F which makes eq.5a) E = 2×D
Hint #2
In eq.2, replace E with 2×D (from eq.5a): B + C + D = 2×D Subtract D from both sides of the above equation: B + C + D - D = 2×D - D which becomes B + C = D Replace D with A + B (from eq.3): B + C = A + B Subtract B from both sides: B + C - B = A + B - B which means C = A
Hint #3
eq.6 may be written as: A + 10×A + B + 10×C + D = 10×D + E Subtract D from both sides of the above equation: A + 10×A + B + 10×C + D - D = 10×D + E - D which becomes 10×A + A + B + 10×C = 9×D + E Substitute D for A + B (from eq.3), A for C, and 2×D for E (from eq.5a) in the above equation: 10×A + D + 10×A = 9×D + 2×D which becomes 20×A + D = 11×D Subtract D from both sides: 20×A + D - D = 11×D - D which becomes 20×A = 10×D Divide both sides by 10: 20×A ÷ 10 = 10×D ÷ 10 which makes 2×A = D
Hint #4
Substitute (2×A) for D in eq.5a: E = 2×(2×A) which makes E = 4×A
Hint #5
Substitute 2×A for D in eq.3: 2×A = A + B Subtract A from both sides: 2×A - A = A + B - A which makes A = B
Hint #6
Add F to both sides of eq.4: A + B + C + D + E - F + F = F + F which becomes A + B + C + D + E = 2×F Substitute 2×F for A + B + C + D + E in eq.1: 2×F + F = 27 which becomes 3×F = 27 Divide both sides by 3: 3×F ÷ 3 = 27 ÷ 3 which makes F = 9
Solution
Substitute A for B and C, 2×A for D, 4×A for E, and 9 for F in eq.1: A + A + A + 2×A + 4×A + 9 = 27 which becomes 9×A + 9 = 27 Subtract 9 from both sides of the above equation: 9×A + 9 - 9 = 27 - 9 which makes 9×A = 18 Divide both sides by 9: 9×A ÷ 9 = 18 ÷ 9 which means A = 2 making B = C = A = 2 D = 2×A = 2 × 2 = 4 E = 4×A = 4 × 2 = 8 and ABCDEF = 222489