Puzzle for August 22, 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
In eq.4, substitute (C + E) for F (from eq.5): C – D = E – (C + E) which is equivalent to C – D = E – C – E which becomes C – D = –C Add C and D to both sides of the equation above: C – D + C + D = –C + C + D which makes 2×C = D
Hint #2
Subtract the left and right sides of eq.6 from the left and right sides of eq.2, respectively: E + F – (D + E) = A + D – (A – D) which is equivalent to E + F – D – E = A + D – A + D which becomes F – D = 2×D Add D to both sides of the above equation: F – D + D = 2×D + D which makes F = 3×D Replace D with (2×C): F = 3×(2×C) which makes F = 6×C
Hint #3
In eq.5, replace F with 6×C: 6×C = C + E Subtract C from both sides of the equation above: 6×C – C = C + E – C which makes 5×C = E
Hint #4
In eq.6, substitute 2×C for D, and 5×C for E: 2×C + 5×C = A – 2×C which becomes 7×C = A – 2×C Add 2×C to both sides of the above equation: 7×C + 2×C = A – 2×C + 2×C which makes 9×C = A
Hint #5
Substitute 2×C for D, and 6×C for F in eq.3: B – C = 2×C + 6×C which becomes B – C = 8×C Add C to both sides of the above equation: B – C + C = 8×C + C which makes B = 9×C
Solution
Substitute 9×C for A and B, 2×C for D, 5×C for E, and 6×C for F in eq.1: 9×C + 9×C + C + 2×C + 5×C + 6×C = 32 which simplifies to 32×C = 32 Divide both sides of the above equation by 32: 32×C ÷ 32 = 32 ÷ 32 which means C = 1 making A = B = 9×C = 9 × 1 = 9 D = 2×C = 2 × 1 = 2 E = 5×C = 5 × 1 = 5 F = 6×C = 6 × 1 = 6 and ABCDEF = 991256