Puzzle for July 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 positive integer.
Scratchpad
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Hint #1
In eq.5, replace E with C × D (from eq.6): D × C × D = C × F Divide both sides of the above equation by C: D × C × D ÷ C = C × F ÷ C which makes eq.5a) D × D = F
Hint #2
In eq.2, replace F with D × D (from eq.5a): D × D = D + E Subtract D from each side of the equation above: D × D - D = D + E - D which means eq.2a) D × (D - 1) = E
Hint #3
In eq.2a, replace E with C × D (from eq.6): D × (D - 1) = C × D Divide both sides of the above equation by D: D × (D - 1) ÷ D = C × D ÷ D which makes eq.2b) D - 1 = C
Hint #4
In eq.4, substitute (D - 1) for C (from eq.2b): A - D = D - (D - 1) Add D to both sides: A - D + D = D - (D - 1) + D which becomes A = 2×D - D + 1 which makes eq.4a) A = D + 1
Hint #5
Substitute D × (D - 1) for E (from eq.2a), D + 1 for A (from eq.4a), and D - 1 for C (from eq.2b) in eq.3: D × (D - 1) = D + 1 + D - 1 which becomes D × (D - 1) = 2×D Divide both sides of the above equation by D: D × (D - 1) ÷ D = 2×D ÷ D which makes D - 1 = 2 Add 1 to both sides: D - 1 + 1 = 2 + 1 which means D = 3
Hint #6
Substitute 3 for D in eq.4a: A = D + 1 = 3 + 1 = 4 Substitute 3 for D in eq.2b: C = D - 1 = 3 - 1 = 2 Substitute 3 for D in eq.2a: E = D × (D - 1) = 3 × (3 - 1) = 3 × 2 = 6 Substitute 3 for D in eq.5a: F = D × D = 3 × 3 = 9
Solution
Substitute 4 for A, 2 for C, 3 for D, 6 for E, and 9 for F in eq.1: 4 + B + 2 + 3 + 6 + 9 = 25 which simplifies to B + 24 = 25 Subtract 24 from both sides of the equation above: B + 24 - 24 = 25 - 24 which means B = 1 making ABCDEF = 412369