Puzzle for May 11, 2023 ( )
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 F with C + E (from eq.2): C + C + E = D + E which becomes 2×C + E = D + E Subtract E from each side of the above equation: 2×C + E - E = D + E - E which makes 2×C = D
Hint #2
In eq.3, replace D with 2×C: B = C + 2×C which makes B = 3×C
Hint #3
In eq.2, substitute A + B for F (from eq.4): eq.2a) A + B = C + E
Hint #4
Substitute 3×C for B in eq.2a: A + 3×C = C + E Subtract C from each side of the equation above: A + 3×C - C = C + E - C which becomes eq.2b) A + 2×C = E
Hint #5
Substitute 3×C for B in eq.4: eq.4a) F = A + 3×C
Hint #6
Substitute A + 2×C for E (from eq.2b), 3×C for B, 2×C for D, and A + 3×C for F (from eq.4a) in eq.6: A + A + 2×C = 3×C + 2×C + A + 3×C which becomes 2×A + 2×C = 8×C + A Subtract 2×C and A from both sides of the equation above: 2×A + 2×C - 2×C - A = 8×C + A - 2×C - A which becomes A = 6×C
Hint #7
Substitute 6×C for A in eq.4a: F = 6×C + 3×C which makes F = 9×C
Hint #8
Substitute 6×C for A in eq.2b: 6×C + 2×C = E which makes 8×C = E
Solution
Substitute 6×C for A, 3×C for B, 2×C for D, 8×C for E, and 9×C for F in eq.1: 6×C + 3×C + C + 2×C + 8×C + 9×C = 29 which simplifies to 29×C = 29 Divide both sides of the above equation by 29: 29×C ÷ 29 = 29 ÷ 29 which means C = 1 making A = 6×C = 6 × 1 = 6 B = 3×C = 3 × 1 = 3 D = 2×C = 2 × 1 = 2 E = 8×C = 8 × 1 = 8 F = 9×C = 9 × 1 = 9 and ABCDEF = 631289