Puzzle for December 27, 2021 ( )
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.5, replace D with A + C (from eq.2): C – A = A + C – C which becomes C – A = A Add A to both sides of the above equation: C – A + A = A + A which makes C = 2×A
Hint #2
In eq.2, replace C with 2×A: D = A + 2×A which makes D = 3×A
Hint #3
In eq.3, substitute 3×A for D: F = A + 3×A which makes F = 4×A
Hint #4
Substitute 3×A for D, and 4×A for F in eq.4: B = 3×A + 4×A which makes B = 7×A
Hint #5
Substitute 3×A for D, 7×A for B, and 2×A for C in eq.6: 3×A + E = A + 7×A + 2×A which becomes 3×A + E = 10×A Subtract 3×A from each side of the above equation: 3×A + E – 3×A = 10×A – 3×A which makes E = 7×A
Solution
Substitute 7×A for B and E, 2×A for C, 3×A for D, and 4×A for F in eq.1: A + 7×A + 2×A + 3×A + 7×A + 4×A = 24 which simplifies to 24×A = 24 Divide both sides of the above equation by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = E = 7×A = 7×1 = 7 C = 2×A = 2×1 = 2 D = 3×A = 3×1 = 3 F = 4×A = 4×1 = 4 and ABCDEF = 172374