Puzzle for December 21, 2020 ( )
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 A + B with D (from eq.2): D + D = C which makes eq.6a) 2×D = C
Hint #2
In eq.4, replace F with A + C (from eq.3): A + A + C = B + C which becomes 2×A + C = B + C Subtract C from both sides of the above equation: 2×A + C – C = B + C – C which makes 2×A = B
Hint #3
In eq.2, substitute 2×A for B: A + 2×A = D which makes 3×A = D
Hint #4
Substitute (3×A) for D in eq.6a: 2×(3×A) = C which makes 6×A = C
Hint #5
Substitute 3×A for D in eq.5: A + 3×A = E which makes 4×A = E
Hint #6
Substitute 6×A for C in eq.3: A + 6×A = F which makes 7×A = F
Solution
Substitute 2×A for B, 6×A for C, 3×A for D, 4×A for E, and 7×A for F in eq.1: A + 2×A + 6×A + 3×A + 4×A + 7×A = 23 which simplifies to 23×A = 23 Divide both sides of the equation above by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 6×A = 6 × 1 = 6 D = 3×A = 3 × 1 = 3 E = 4×A = 4 × 1 = 4 F = 7×A = 7 × 1 = 7 and ABCDEF = 126347