Puzzle for August 25, 2020 ( )
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
Help Area
Hint #1
In eq.4, replace E with A + C (from eq.6): A – C = A + C – F Subtract A from both sides of the above equation: A – C – A = A + C – F – A which becomes –C = C – F Add C and F to both sides: –C + C + F = C – F + C + F which makes F = 2×C
Hint #2
In eq.5, replace E with A + C (from eq.6): B + C + A + C = A + D which becomes B + 2×C + A = A + D Subtract A from both sides of the equation above: B + 2×C + A – A = A + D – A which becomes eq.5a) B + 2×C = D
Hint #3
In eq.2, substitute B + 2×C for D (from eq.5a), and 2×C for F: B + 2×C + 2×C = A + B which becomes B + 4×C = A + B Subtract B from each side of the equation above: B + 4×C – B = A + B – B which makes 4×C = A
Hint #4
Substitute 4×C for A in eq.6: E = 4×C + C which makes E = 5×C
Hint #5
Substitute 5×C for E, and 2×C for F in eq.3: 5×C + 2×C = D – C which becomes 7×C = D – C Add C to each side of the above equation: 7×C + C = D – C + C which makes 8×C = D
Hint #6
Substitute 8×C for D in eq.5a: B + 2×C = 8×C Subtract 2×C from both sides of the equation above: B + 2×C – 2×C = 8×C – 2×C which makes B = 6×C
Solution
Substitute 4×C for A, 6×C for B, 8×C for D, 5×C for E, and 2×C for F in eq.1: 4×C + 6×C + C + 8×C + 5×C + 2×C = 26 which simplifies to 26×C = 26 Divide both sides of the above equation by 26: 26×C ÷ 26 = 26 ÷ 26 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 6×C = 6 × 1 = 6 D = 8×C = 8 × 1 = 8 E = 5×C = 5 × 1 = 5 F = 2×C = 2 × 1 = 2 and ABCDEF = 461852