Puzzle for November 2, 2020 ( )
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 B × C (from eq.4): A × B × C = C Divide both sides of the above equation by C: A × B × C ÷ C = C ÷ C which becomes A × B = 1 Since A and B must be positive integers, the above equation makes: A = 1 and B = 1
Hint #2
In eq.5, replace A with 1: 1 × E = C which makes E = C
Hint #3
In eq.3, substitute 1 for B: eq.3a) D = 1 + F
Hint #4
Substitute 1 + F for D (from eq.3a) in eq.2: E = 1 + F + F which makes E = 1 + 2×F and also makes eq.2a) C = E = 1 + 2×F
Solution
Substitute 1 for A and B, 1 + 2×F for C and E (from eq.2a), and 1 + F for D (from eq.3a) in eq.1: 1 + 1 + 1 + 2×F + 1 + F + 1 + 2×F + F = 29 which simplifies to 5 + 6×F = 29 Subtract 5 from both sides of the above equation: 5 + 6×F – 5 = 29 – 5 which makes 6×F = 24 Divide both sides by 6: 6×F ÷ 6 = 24 ÷ 6 which means F = 4 making C = E = 1 + 2×F = 1 + 2×4 = 1 + 8 = 9 (from eq.2a) D = 1 + F = 1 + 4 = 5 (from eq.3a) and ABCDEF = 119594