Puzzle for May 11, 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.2, replace B with C + D (from eq.3): D + E = C + D + C which becomes D + E = 2×C + D Subtract D from each side of the equation above: D + E – D = 2×C + D – D which makes E = 2×C
Hint #2
In eq.4, replace E with 2×C: C + 2×C = A – 2×C which becomes 3×C = A – 2×C Add 2×C to each side of the above equation: 3×C + 2×C = A – 2×C + 2×C which makes 5×C = A
Hint #3
In eq.6, substitute 5×C for A: F = 5×C + C which makes F = 6×C
Hint #4
Substitute 5×C for A, 6×C for F, and 2×C for E in eq.5: 5×C + 6×C = B + 2×C which becomes 11×C = B + 2×C Subtract 2×C from both sides of the above equation: 11×C – 2×C = B + 2×C – 2×C which makes 9×C = B
Hint #5
Substitute 9×C for B in eq.3: 9×C = C + D Subtract C from both sides of the equation above: 9×C – C = C + D – C which makes 8×C = D
Solution
Substitute 5×C for A, 9×C for B, 8×C for D, 2×C for E, and 6×C for F in eq.1: 5×C + 9×C + C + 8×C + 2×C + 6×C = 31 which simplifies to 31×C = 31 Divide both sides of the above equation by 31: 31×C ÷ 31 = 31 ÷ 31 which means C = 1 making A = 5×C = 5 × 1 = 5 B = 9×C = 9 × 1 = 9 D = 8×C = 8 × 1 = 8 E = 2×C = 2 × 1 = 2 F = 6×C = 6 × 1 = 6 and ABCDEF = 591826