Puzzle for April 24, 2023 ( )
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.
Scratchpad
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Hint #1
In eq.4, substitute (D + E) for B (from eq.3): D - E = E - (D + E) which becomes D - E = E - D - E which becomes D - E = -D Add E and D to both sides of the above equation: D - E + E + D = -D + E + D which makes eq.4a) 2×D = E
Hint #2
In eq.5, replace E with 2×D: 2×D - C = C + D In the above equation, add C to both sides, and subtract D from both sides: 2×D - C + C - D = C + D + C - D which makes D = 2×C
Hint #3
In eq.4a, replace D with (2×C): 2×(2×C) = E which makes 4×C = E
Hint #4
In eq.3, substitute 2×C for D, and 4×C for E: B = 2×C + 4×C which makes B = 6×C
Hint #5
Substitute 6×C for B in eq.2: 6×C = A + C Subtract C from each side of the equation above: 6×C - C = A + C - C which makes 5×C = A
Hint #6
Substitute 5×C for A, and 4×C for E in eq.6: 5×C + 4×C = C + F which becomes 9×C = C + F Subtract C from both sides of the above equation: 9×C - C = C + F - C which makes 8×C = F
Solution
Substitute 5×C for A, 6×C for B, 2×C for D, 4×C for E, and 8×C for F in eq.1: 5×C + 6×C + C + 2×C + 4×C + 8×C = 26 which simplifies to 26×C = 26 Divide both sides of the above equation by 26: 26×C ÷ 26 = 26 ÷ 26 which means C = 1 making A = 5×C = 5 × 1 = 5 B = 6×C = 6 × 1 = 6 D = 2×C = 2 × 1 = 2 E = 4×C = 4 × 1 = 4 F = 8×C = 8 × 1 = 8 and ABCDEF = 561248