Puzzle for August 19, 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 F with B + C (from eq.4): B + D = C + B + C which becomes B + D = B + 2×C Subtract B from both sides of the above equation: B + D – B = B + 2×C – B which makes D = 2×C
Hint #2
In eq.5, replace D with 2×C: 2×C = A + C Subtract C from each side of the equation above: 2×C – C = A + C – C which makes C = A
Hint #3
In eq.6, substitute 2×C for D: E = C + 2×C which makes E = 3×C
Hint #4
Substitute 2×C for D, 3×C for E, and C for A in eq.2: 2×C + 3×C = C + B + C which becomes 5×C = B + 2×C Subtract 2×C from each side of the above equation: 5×C – 2×C = B + 2×C – 2×C which makes 3×C = B
Hint #5
Substitute 3×C for B in eq.4: F = 3×C + C which means F = 4×C
Solution
Substitute C for A, 3×C for B and E, 2×C for D, and 4×C for F in eq.1: C + 3×C + C + 2×C + 3×C + 4×C = 28 which simplifies to 14×C = 28 Divide both sides of the equation above by 14: 14×C ÷ 14 = 28 ÷ 14 which means C = 2 making A = C = 2 B = E = 3×C = 3 × 2 = 6 D = 2×C = 2 × 2 = 4 F = 4×C = 4 × 2 = 8 and ABCDEF = 262468