Puzzle for June 29, 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.4, replace E with B + D (from eq.2), and 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 equation above: B + D – D = 2×A + D – D which becomes B = 2×A
Hint #2
In eq.6, replace B with 2×A: C – 2×A = 2×A + F Add 2×A to both sides of the above equation: C – 2×A + 2×A = 2×A + F + 2×A which becomes eq.6a) C = 4×A + F
Hint #3
In eq.5, substitute 2×A for B, 4×A + F for C (from eq.6a), and A + F for E (from eq.4): 2×A + 4×A + F = A + F + F which becomes 6×A + F = A + 2×F Subtract F and A from each side of the equation above: 6×A + F – F – A = A + 2×F – F – A which makes 5×A = F
Hint #4
Substitute 5×A for F in eq.4: E = A + 5×A which makes E = 6×A
Hint #5
Substitute 6×A for E, and 2×A for B in eq.2: 6×A = 2×A + D Subtract 2×A from each side of the above equation: 6×A – 2×A = 2×A + D – 2×A which makes 4×A = D
Hint #6
Substitute 2×A for B, and 5×A for F in eq.6: C – 2×A = 2×A + 5×A which becomes C – 2×A = 7×A Add 2×A to both sides of the equation above: C – 2×A + 2×A = 7×A + 2×A which makes C = 9×A
Solution
Substitute 2×A for B, 9×A for C, 4×A for D, 6×A for E, and 5×A for F in eq.1: A + 2×A + 9×A + 4×A + 6×A + 5×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 9×A = 9 × 1 = 9 D = 4×A = 4 × 1 = 4 E = 6×A = 6 × 1 = 6 F = 5×A = 5 × 1 = 5 and ABCDEF = 129465