Puzzle for June 22, 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
Subtract C from each side of eq.4: B – C + D – C = C + F – C which becomes eq.4a) B – 2×C + D = F In eq.6, replace F with B – 2×C + D (from eq.4a): E + B – 2×C + D – C = A + B + C which becomes E + B – 3×C + D = A + B + C In the equation above, subtract B from both sides, and add 3×C to both sides: E + B – 3×C + D – B + 3×C = A + B + C – B + 3×C which becomes E + D = A + 4×C which is the same as eq.6a) D + E = A + 4×C
Hint #2
In eq.3, replace D + E with A + 4×C (from eq.6a): A + C = A + 4×C Subtract A and C from each side of the above equation: A + C – A – C = A + 4×C – A – C which becomes 0 = 3×C which means 0 = C
Hint #3
In eq.3, substitute 0 for C: D + E = A + 0 which becomes eq.3a) D + E = A
Hint #4
Substitute D + E for A (from eq.3a), and 0 for C in eq.5: D + E – 0 + D = B + E which becomes 2×D + E = B + E Subtract E from each side of the above equation: 2×D + E – E = B + E – E which makes 2×D = B
Hint #5
Substitute 2×D for B, and 0 for C in eq.4a: 2×D – 2×0 + D = F which makes 3×D = F
Hint #6
Substitute 0 for C, 2×D for B, and 3×D for F in eq.2: 0 + E = 2×D + 3×D which makes E = 5×D
Hint #7
Substitute 5×D for E, and 0 for C in eq.3: D + 5×D = A + 0 which makes 6×D = A
Solution
Substitute 6×D for A, 2×D for B, 0 for C, 5×D for E, and 3×D for F in eq.1: 6×D + 2×D + 0 + D + 5×D + 3×D = 17 which simplifies to 17×D = 17 Divide both sides of the above equation by 17: 17×D ÷ 17 = 17 ÷ 17 which means D = 1 making A = 6×D = 6 × 1 = 6 B = 2×D = 2 × 1 = 2 E = 5×D = 5 × 1 = 5 F = 3×D = 3 × 1 = 3 and ABCDEF = 620153