Puzzle for March 29, 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 and DE are 2-digit numbers (not A×B or D×E).
Scratchpad
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Hint #1
Add C + D to both sides of eq.5: F - C + C + D = C - D + C + D which becomes eq.5a) F + D = 2×C
Hint #2
eq.3 may be written as: B + D + C = E In the above equation, replace B + D with A (from eq.2): eq.4a) A + C = E
Hint #3
In eq.4, replace E with A + C (from eq.4a): C + A + C = A + B + F Subtract A from each side of the above equation: C + A + C - A = A + B + F - A which becomes eq.4b) 2×C = B + F
Hint #4
In eq.4b, substitute F + D for 2×C (from eq.5a): F + D = B + F Subtract F from each side: F + D - F = B + F - F which makes D = B
Hint #5
Substitute B for D in eq.2: A = B + B which makes A = 2×B
Hint #6
eq.6 may be written as: 10×A + B + C = 10×D + E + F Substitute B + C + D for E (from eq.3) in the equation above: 10×A + B + C = 10×D + B + C + D + F Subtract both B and C from each side: 10×A + B + C - B - C = 10×D + B + C + D + F - B - C which simplifies to eq.6a) 10×A = 11×D + F
Hint #7
Substitute (2×B) for A, and B for D in eq.6a: 10×(2×B) = 11×B + F which is equivalent to 20×B = 11×B + F Subtract 11×B from both sides: 20×B - 11×B = 11×B + F - 11×B which means 9×B = F
Hint #8
Substitute B for D, and 9×B for F in eq.5a: 2×C = B + 9×B which becomes 2×C = 10×B Divide both sides of the above equation by 2: 2×C ÷ 2 = 10×B ÷ 2 which makes C = 5×B
Hint #9
Substitute B for D, and 5×B for C in eq.3: B + 5×B + B = E which makes 7×B = E
Solution
Substitute 2×B for A, 5×B for C, B for D, 7×B for E, and 9×B for F in eq.1: 2×B + B + 5×B + B + 7×B + 9×B = 25 which simplifies to 25×B = 25 Divide both sides by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 5×B = 5 × 1 = 5 D = B = 1 E = 7×B = 7 × 1 = 7 F = 9×B = 9 × 1 = 9 and ABCDEF = 215179