Puzzle for July 13, 2019 ( )
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
eq.4 may be written as: C - B = D - (A + E) In the above equation, replace A + E with B + C (from eq.2): C - B = D - (B + C) which is the same as C - B = D - B - C Add B and C to both sides: C - B + B + C = D - B - C + B + C which simplifies to eq.4a) 2×C = D
Hint #2
In eq.6, replace D with 2×C: 2×C - C + E = B - E which becomes C + E = B - E Express B in terms of C and E by adding E to both sides of the equation above: C + E + E = B - E + E which becomes eq.6a) C + 2×E = B
Hint #3
In eq.2, substitute C + 2×E for B (from eq.6a): C + 2×E + C = A + E which becomes 2×C + 2×E = A + E Express A in terms of C and E by subtracting E from each side of the above equation: 2×C + 2×E - E = A + E - E which becomes eq.2a) 2×C + E = A
Hint #4
Express F in terms of C and E by substituting C + 2×E for B (from eq.6a) in eq.3: F = C + 2×E + E which becomes eq.3a) F = C + 3×E
Hint #5
Solve for C and E by substituting 2×C + E for A (from eq.2a), 2×C for D, C + 2×E for B (from eq.6a), and C + 3×E for F (from eq.3a) in eq.5: 2×C + E - C + 2×C = C + 2×E + C + 3×E which becomes 3×C + E = 2×C + 5×E Subtract both 2×C and E from each side of the above equation: 3×C + E - 2×C - E = 2×C + 5×E - 2×C - E which simplifies to C = 4×E
Hint #6
Substitute (4×E) for C in eq.2a: 2×(4×E) + E = A which becomes 8×E + E = A which makes 9×E = A
Hint #7
Substitute 4×E for C in eq.6a: 4×E + 2×E = B which makes 6×E = B
Hint #8
Substitute (4×E) for C in eq.4a: 2×(4×E) = D which makes 8×E = D
Hint #9
Substitute 4×E for C in eq.3a: F = 4×E + 3×E which makes F = 7×E
Solution
Substitute 9×E for A, 6×E for B, 4×E for C, 8×E for D, and 7×E for F in eq.1: 9×E + 6×E + 4×E + 8×E + E + 7×E = 35 which simplifies to 35×E = 35 Divide both sides by 35: 35×E ÷ 35 = 35 ÷ 35 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 6×E = 6 × 1 = 6 C = 4×E = 4 × 1 = 4 D = 8×E = 8 × 1 = 8 F = 7×E = 7 × 1 = 7 and ABCDEF = 964817