Puzzle for August 28, 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.6, replace F with C + E (from eq.4): E - C = C + E - E which becomes E - C = C Add C to both sides of the above equation: E - C + C = C + C which makes E = 2×C
Hint #2
In eq.4, substitute 2×C for E: F = C + 2×C which makes F = 3×C
Hint #3
In eq.5, substitute 2×C for E, and 3×C for F: A = 2×C + 3×C which makes A = 5×C
Hint #4
Substitute 5×C for A, and 3×C for F in eq.2: B = 5×C + 3×C which makes B = 8×C
Hint #5
Substitute 5×C for A, and 2×C for E in eq.3: D = 5×C + 2×C which makes D = 7×C
Solution
Substitute 5×C for A, 8×C for B, 7×C for D, 2×C for E, and 3×C for F in eq.1: 5×C + 8×C + C + 7×C + 2×C + 3×C = 26 which simplifies to 26×C = 26 Divide both sides of the above equation by 26: 26×C ÷ 26 = 26 ÷ 26 which means C = 1 making A = 5×C = 5 × C = 5 B = 8×C = 8 × C = 8 D = 7×C = 7 × C = 7 E = 2×C = 2 × C = 2 F = 3×C = 3 × C = 3 and ABCDEF = 581723