Puzzle for November 18, 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.4, replace D with A + B (from eq.2): E = A + A + B which becomes eq.4a) E = 2×A + B
Hint #2
In eq.6, replace E with 2×A + B (from eq.4a): B + C = A + 2×A + B which becomes B + C = 3×A + B Subtract B from each side of the above equation: B + C – B = 3×A + B – B which makes C = 3×A
Hint #3
In eq.5, substitute C + D for F (from eq.3): C + C + D = B + E which becomes eq.5a) 2×C + D = B + E
Hint #4
Substitute (3×A) for C, and 2×A + B for E (from eq.4a) in eq.5a: 2×(3×A) + D = B + 2×A + B which becomes 6×A + D = 2×B + 2×A Subtract 2×A from both sides of the above equation: 6×A + D – 2×A = 2×B + 2×A – 2×A which becomes eq.5b) 4×A + D = 2×B
Hint #5
Substitute A + B for D (from eq.2) in eq.5b: 4×A + A + B = 2×B which becomes 5×A + B = 2×B Subtract B from each side of the above equation: 5×A + B – B = 2×B – B which makes 5×A = B
Hint #6
Substitute 5×A for B in eq.4a: E = 2×A + 5×A which makes E = 7×A
Hint #7
Substitute 5×A for B in eq.2: D = A + 5×A which makes D = 6×A
Hint #8
Substitute 3×A for C, and 6×A for D in eq.3: F = 3×A + 6×A which makes F = 9×A
Solution
Substitute 5×A for B, 3×A for C, 6×A for D, 7×A for E, and 9×A for F in eq.1: A + 5×A + 3×A + 6×A + 7×A + 9×A = 31 which simplifies to 31×A = 31 Divide both sides of the above equation by 31: 31×A ÷ 31 = 31 ÷ 31 which means A = 1 making B = 5×A = 5 × 1 = 5 C = 3×A = 3 × 1 = 3 D = 6×A = 6 × 1 = 6 E = 7×A = 7 × 1 = 7 F = 9×A = 9 × 1 = 9 and ABCDEF = 153679