Puzzle for November 17, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.6, replace B with A + C (from eq.4): A + C + C = A – C + F which becomes A + 2×C = A – C + F In the above equation, subtract A from both sides, and add C to both sides: A + 2×C – A + C = A – C + F – A + C which makes 3×C = F
Hint #2
In eq.3, replace F with 3×C: E + 3×C = B + C Subtract C from both sides of the above equation: E + 3×C – C = B + C – C which becomes eq.3a) E + 2×C = B
Hint #3
In eq.5, substitute E + 2×C for B (from eq.3a), and 3×C for F: D + E = E + 2×C + C + 3×C which becomes D + E = E + 6×C Subtract E from each side of the above equation: D + E – E = E + 6×C – E which makes D = 6×C
Hint #4
Substitute E + 2×C for B (from eq.3a), and 6×C for D in eq.2: E + 2×C + 6×C = A + E which becomes E + 8×C = A + E Subtract E from each side of the equation above: E + 8×C – E = A + E – E which makes 8×C = A
Hint #5
Substitute 8×C for A in eq.4: 8×C + C = B which makes 9×C = B
Hint #6
Substitute 9×C for B in eq.3a: E + 2×C = 9×C Subtract 2×C from both sides of the equation above: E + 2×C – 2×C = 9×C – 2×C which makes E = 7×C
Solution
Substitute 8×C for A, 9×C for B, 6×C for D, 7×C for E, and 3×C for F in eq.1: 8×C + 9×C + C + 6×C + 7×C + 3×C = 34 which simplifies to 34×C = 34 Divide both sides of the equation above by 34: 34×C ÷ 34 = 34 ÷ 34 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 9×C = 9 × 1 = 9 D = 6×C = 6 × 1 = 6 E = 7×C = 7 × 1 = 7 F = 3×C = 3 × 1 = 3 and ABCDEF = 891673