Puzzle for July 5, 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 A with C + D (from eq.2), and F with C + E (from eq.5): C + D + D = C + E which becomes C + 2×D = C + E Subtract C from each side of the equation above: C + 2×D – C = C + E – C which makes 2×D = E
Hint #2
In eq.6, replace F with C + E (from eq.5): B + C = D + E + C + E which becomes B + C = D + 2×E + C Subtract C from each side of the above equation: B + C – C = D + 2×E + C – C which becomes eq.6a) B = D + 2×E
Hint #3
In eq.6a, substitute (2×D) for E: B = D + 2×(2×D) which becomes B = D + 4×D which makes B = 5×D
Hint #4
Substitute 5×D for B, and 2×D for E in eq.3: 5×D + 2×D = C which makes 7×D = C
Hint #5
Substitute 7×D for C in eq.2: 7×D + D = A which makes 8×D = A
Hint #6
Substitute 8×D for A in eq.4: 8×D + D = F which makes 9×D = F
Solution
Substitute 8×D for A, 5×D for B, 7×D for C, 2×D for E, and 9×D for F in eq.1: 8×D + 5×D + 7×D + D + 2×D + 9×D = 32 which simplifies to 32×D = 32 Divide both sides of the above equation by 32: 32×D ÷ 32 = 32 ÷ 32 which means D = 1 making A = 8×D = 8 × 1 = 8 B = 5×D = 5 × 1 = 5 C = 7×D = 7 × 1 = 7 E = 2×D = 2 × 1 = 2 F = 9×D = 9 × 1 = 9 and ABCDEF = 857129