Puzzle for March 25, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.3, replace B with C + F (from eq.6): C + F + C = D + F which becomes 2×C + F = D + F Subtract F from both sides of the above equation: 2×C + F - F = D + F - F which makes eq.3a) 2×C = D
Hint #2
In eq.3, replace F with C + D (from eq.4): B + C = D + C + D which becomes B + C = 2×D + C Subtract C from each side of the above equation: B + C - C = 2×D + C - C which makes B = 2×D In the above equation, substitute 2×C for D (from eq.3a): B = 2×2×C which makes B = 4×C
Hint #3
Substitute 4×C for B, and 2×C for D in eq.3: 4×C + C = 2×C + F which becomes 5×C = 2×C + F Subtract 2×C from each side of the above equation: 5×C - 2×C = 2×C + F - 2×C which means 3×C = F
Hint #4
Substitute 4×C for B, and 2×C for D in eq.5: E = 4×C + 2×C which makes E = 6×C
Hint #5
Substitute 6×C for E, and 3×C for F in eq.2: A + C = 6×C + 3×C which becomes A + C = 9×C Subtract C from both sides: A + C - C = 9×C - C which makes A = 8×C
Solution
Substitute 8×C for A, 4×C for B, 2×C for D, 6×C for E, and 3×C for F in eq.1: 8×C + 4×C + C + 2×C + 6×C + 3×C = 24 which simplifies to 24×C = 24 Divide both sides by 24: 24×C ÷ 24 = 24 ÷ 24 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 4×C = 4 × 1 = 4 D = 2×C = 2 × 1 = 2 E = 6×C = 6 × 1 = 6 F = 3×C = 3 × 1 = 3 and ABCDEF = 841263