Puzzle for May 9, 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.3, replace C with A + B (from eq.2): D = A + A + B which becomes eq.3a) D = 2×A + B
Hint #2
In eq.4, replace D with 2×A + B (from eq.3a): F = A + 2×A + B which becomes eq.4a) F = 3×A + B
Hint #3
In eq.5, substitute 3×A + B for F (from eq.4a): A - B = B - (3×A + B) which becomes A - B = B - 3×A - B which becomes A - B = -3×A Add B and 3×A to both sides of the above equation: A - B + B + 3×A = -3×A + B + 3×A which simplifies to 4×A = B
Hint #4
Substitute 4×A for B in eq.4a: F = 3×A + 4×A which makes F = 7×A
Hint #5
Substitute 4×A for B in eq.3a: D = 2×A + 4×A which makes D = 6×A
Hint #6
Substitute 4×A for B in eq.2: C = A + 4×A which makes C = 5×A
Hint #7
Substitute 4×A for B, 7×A for F, and 5×A for C in eq.6: 4×A + 7×A = 5×A + E which becomes 11×A = 5×A + E Subtract 5×A from each side of the equation above: 11×A - 5×A = 5×A + E - 5×A which makes 6×A = E
Solution
Substitute 4×A for B, 5×A for C, 6×A for D and E, and 7×A for F in eq.1: A + 4×A + 5×A + 6×A + 6×A + 7×A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = 4×A = 4 × 1 = 4 C = 5×A = 5 × 1 = 5 D = E = 6×A = 6 × 1 = 6 F = 7×A = 7 × 1 = 7 and ABCDEF = 145667