Puzzle for July 26, 2021 ( )
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.5, replace E with C + D (from eq.2): D + F = C + C + D which becomes D + F = 2×C + D Subtract D from each side of the above equation: D + F – D = 2×C + D – D which makes F = 2×C
Hint #2
In eq.6, replace E with C + D (from eq.2), and F with 2×C: C + D + 2×C = A + D which becomes 3×C + D = A + D Subtract D from each side of the above equation: 3×C + D – D = A + D – D which makes 3×C = A
Hint #3
In eq.3, substitute 3×C for A, and 2×C for F: E = 3×C + 2×C which makes E = 5×C
Hint #4
Substitute 5×C for E in eq.2: C + D = 5×C Subtract C from both sides of the equation above: C + D – C = 5×C – C which becomes D = 4×C
Hint #5
Substitute 4×C for D, 3×C for A, and 5×C for E in eq.4: B + 4×C = 3×C + 5×C which becomes B + 4×C = 8×C Subtract 4×C from both sides of the equation above: B + 4×C – 4×C = 8×C – 4×C which makes B = 4×C
Solution
Substitute 3×C for A, 4×C for B and D, 5×C for E, and 2×C for F in eq.1: 3×C + 4×C + C + 4×C + 5×C + 2×C = 19 which simplifies to 19×C = 19 Divide both sides of the above equation by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = 3×C = 3 × 1 = 3 B = D = 4×C = 4 × 1 = 4 E = 5×C = 5 × 1 = 5 F = 2×C = 2 × 1 = 2 and ABCDEF = 341452