Puzzle for March 28, 2022 ( )
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 C with A + D (from eq.4): A + D + D = A + F which becomes A + 2×D = A + F Subtract A from each side of the above equation: A + 2×D – A = A + F – A which makes 2×D = F
Hint #2
In eq.2, replace F with 2×D: A = D + 2×D which makes A = 3×D
Hint #3
In eq.4, substitute 3×D for A: C = 3×D + D which makes C = 4×D
Hint #4
Substitute 4×D for C, and 2×D for F in eq.5: E = 4×D + 2×D which makes E = 6×D
Hint #5
Substitute 6×D for E in eq.3: B = D + 6×D which makes B = 7×D
Solution
Substitute 3×D for A, 7×D for B, 4×D for C, 6×D for E, and 2×D for F in eq.1: 3×D + 7×D + 4×D + D + 6×D + 2×D = 23 which simplifies to 23×D = 23 Divide both sides of the above equation by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = 3×D = 3×1 = 3 B = 7×D = 7×1 = 7 C = 4×D = 4×1 = 4 E = 6×D = 6×1 = 6 F = 2×D = 2×1 = 2 and ABCDEF = 374162