Puzzle for May 18, 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
Add A to both sides of eq.5: C - A + A = A + B + A which becomes eq.5a) C = 2×A + B
Hint #2
In eq.4, replace C with 2×A + B (from eq.5a): A + 2×A + B = B + E which becomes 3×A + B = B + E Subtract B from each side of the equation above: 3×A + B - B = B + E - B which makes 3×A = E
Hint #3
In eq.6, substitute B + E for F (from eq.2): E + B + E = C + D - E which becomes 2×E + B = C + D - E Add E to both sides of the above equation: 2×E + B + E = C + D - E + E which becomes eq.6a) 3×E + B = C + D
Hint #4
Subtract the left and right sides of eq.3 from the left and right sides of eq.6a, respectively: 3×E + B - (B + C) = C + D - (D + E) which becomes 3×E + B - B - C = C + D - D - E which becomes 3×E - C = C - E Add C and E to both sides of the above equation: 3×E - C + C + E = C - E + C + E which makes 4×E = 2×C Divide both sides by 2: 4×E ÷ 2 = 2×C ÷ 2 which makes eq.6b) 2×E = C
Hint #5
Substitute 3×A for E in eq.6b: 2×(3×A) = C which makes 6×A = C
Hint #6
Substitute 6×A for C in eq.5a: 6×A = 2×A + B Subtract 2×A from both sides of the above equation: 6×A - 2×A = 2×A + B - 2×A which makes 4×A = B
Hint #7
Substitute 4×A for B, and 3×A for E in eq.2: F = 4×A + 3×A which makes F = 7×A
Hint #8
Substitute 4×A for B, 6×A for C, and 3×A for E in eq.3: 4×A + 6×A = D + 3×A which becomes 10×A = D + 3×A Subtract 3×A from each side of the above equation: 10×A - 3×A = D + 3×A - 3×A which makes 7×A = D
Solution
Substitute 4×A for B, 6×A for C, 7×A for D and F, and 3×A for E in eq.1: A + 4×A + 6×A + 7×A + 3×A + 7×A = 28 which simplifies to 28×A = 28 Divide both sides of the above equation by 28: 28×A ÷ 28 = 28 ÷ 28 which means A = 1 making B = 4×A = 4 × 1 = 4 C = 6×A = 6 × 1 = 6 D = F = 7×A = 7 × 1 = 7 E = 3×A = 3 × 1 = 3 and ABCDEF = 146737