Puzzle for February 6, 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 E with A + D (from eq.5): B + D = A + A + D which becomes B + D = 2×A + D Subtract D from each side of the equation above: B + D - D = 2×A + D - D which makes B = 2×A
Hint #2
In eq.3, replace B with 2×A: C = A + 2×A which makes C = 3×A
Hint #3
In eq.2, substitute 2×A for B, and 3×A for C: F = 2×A + 3×A which makes F = 5×A
Hint #4
Substitute 2×A for B, and 5×A for F in eq.4: D = 2×A + 5×A which makes D = 7×A
Hint #5
Substitute 7×A for D in eq.5: E = A + 7×A which makes E = 8×A
Solution
Substitute 2×A for B, 3×A for C, 7×A for D, 8×A for E, and 5×A for F in eq.1: A + 2×A + 3×A + 7×A + 8×A + 5×A = 26 which simplifies to 26×A = 26 Divide both sides of the above equation by 26: 26×A ÷ 26 = 26 ÷ 26 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 3×A = 3 × 1 = 3 D = 7×A = 7 × 1 = 7 E = 8×A = 8 × 1 = 8 F = 5×A = 5 × 1 = 5 and ABCDEF = 123785