Puzzle for October 17, 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 F with A + D (from eq.3): B + D = A + A + D which becomes B + D = 2×A + D Subtract D from each side of the above equation: B + D - D = 2×A + D - D which makes B = 2×A
Hint #2
In eq.2, replace B with 2×A: 2×A = A + C Subtract A from each side of the equation above: 2×A - A = A + C - A which makes A = C
Hint #3
In eq.4, substitute 2×A for B, and A for C: D = A + 2×A + A which makes D = 4×A
Hint #4
Substitute 4×A for D in eq.3: F = A + 4×A which makes F = 5×A
Hint #5
Substitute 2×A for B, A for C, and 4×A for D in eq.5: E = A + 2×A + A + 4×A which makes E = 8×A
Solution
Substitute 2×A for B, A for C, 4×A for D, 8×A for E, and 5×A for F in eq.1: A + 2×A + A + 4×A + 8×A + 5×A = 21 which simplifies to 21×A = 21 Divide both sides of the above equation by 21: 21×A ÷ 21 = 21 ÷ 21 which means A = 1 making B = 2×A = 2 × 1 = 2 C = A = 1 D = 4×A = 4 × 1 = 4 E = 8×A = 8 × 1 = 8 F = 5×A = 5 × 1 = 5 and ABCDEF = 121485