Puzzle for February 10, 2021 ( )
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 F and A from each side of eq.2: C + F – F – A = A + B – F – A which becomes eq.2a) C – A = B – F Subtract D and F from each side of eq.1: B + D – D – F = A + F – D – F which becomes eq.1a) B – F = A – D
Hint #2
In eq.1a, replace B – F with C – A (from eq.2a): C – A = A – D Add A and D to both sides of the equation above: C – A + A + D = A – D + A + D which becomes eq.1b) C + D = 2×A
Hint #3
In eq.6, replace C + D with 2×A (from eq.1b): B = (2×A) ÷ A which makes B = 2
Hint #4
In eq.4, substitute 2 for B: A – 2 = 2 + C Add 2 to both sides of the equation above: A – 2 + 2 = 2 + C + 2 which makes eq.4a) A = 4 + C
Hint #5
Substitute 4 + C for A (from eq.4a), and 2 for B in eq.2: C + F = 4 + C + 2 which becomes C + F = 6 + C Subtract C from each side of the above equation: C + F – C = 6 + C – C which makes F = 6
Hint #6
Substitute 2 for B, 4 + C for A (from eq.4a), and 6 for F in eq.1: 2 + D = 4 + C + 6 which becomes 2 + D = 10 + C Subtract 2 from each side of the above equation: 2 + D – 2 = 10 + C – 2 which makes eq.1c) D = 8 + C
Hint #7
Substitute 8 + C for D (from eq.1c), 2 for B, and 6 for F in eq.3: 8 + C – E = 2 + 6 which becomes 8 + C – E = 8 In the above equation, subtract 8 from both sides, and add E to both sides: 8 + C – E – 8 + E = 8 – 8 + E which simplifies to C = E
Solution
Substitute C for E, 6 for F, and 4 + C for A (from eq.4a) in eq.5: C × 6 = 4 + C + C which becomes 6×C = 4 + 2×C Subtract 2×C from each side of the above equation: 6×C – 2×C = 4 + 2×C – 2×C which makes 4×C = 4 Divide both sides by 4: 4×C ÷ 4 = 4 ÷ 4 which makes C = 1 making A = 4 + C = 4 + 1 = 5 (from eq.4a) D = 8 + C = 8 + 1 = 9 (from eq.1c) E = C = 1 and ABCDEF = 521916