Puzzle for April 17, 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
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Hint #1
Add A to both sides of eq.2: B – A + A = C + D + A which becomes B = C + D + A In eq.5, replace B with C + D + A: C + D + A – D – F = C + D which becomes C + A – F = C + D In the equation above, subtract C from each side, and add F to each side: C + A – F – C + F = C + D – C + F which becomes eq.5a) A = D + F
Hint #2
In eq.3, replace A with D + F (from eq.5a): D + E = D + F – F which becomes D + E = D Subtract D from both sides of the above equation: D + E – D = D – D which means E = 0
Hint #3
In eq.4, substitute 0 for E: C – 0 + F = A + 0 which becomes eq.4a) C + F = A
Hint #4
Substitute C + F for A (from eq.4a) in eq.5a: C + F = D + F Subtract F from both sides of the equation above: C + F – F = D + F – F which makes C = D
Hint #5
Substitute C for D in eq.5: B – C – F = C + C Add C and F to both sides of the above equation: B – C – F + C + F = C + C + C + F which becomes eq.5a) B = 3×C + F
Hint #6
In eq.6, substitute C + F for A (from eq.4a), and C for D: C + F – C + F = B + C + C which becomes eq.6a) 2×F = B + 2×C
Hint #7
Substitute 3×C + F for B (from eq.5a) in eq.6a: 2×F = 3×C + F + 2×C which becomes 2×F = 5×C + F Subtract F from both sides of the equation above: 2×F – F = 5×C + F – F which makes F = 5×C
Hint #8
Substitute 5×C for F in eq.5a: B = 3×C + 5×C which makes B = 8×C
Hint #9
Substitute 5×C for F in eq.4a: C + 5×C = A which makes 6×C = A
Solution
Substitute 6×C for A, 8×C for B, C for D, 0 for E, and 5×C for F in eq.1: 6×C + 8×C + C + C + 0 + 5×C = 21 which simplifies to 21×C = 21 Divide both sides of the above equation by 21: 21×C ÷ 21 = 21 ÷ 21 which means C = 1 making A = 6×C = 6 × 1 = 6 B = 8×C = 8 × 1 = 8 D = C = 1 F = 5×C = 5 × 1 = 5 and ABCDEF = 681105