Puzzle for February 9, 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
In eq.2, replace D with B + C (from eq.1): B + B + C = A + C which becomes 2×B + C = A + C Subtract C from each side of the above equation: 2×B + C – C = A + C – C which makes 2×B = A
Hint #2
In eq.6, replace A with 2×B: E = 2×B ÷ B which makes E = 2
Hint #3
In eq.5, substitute 2 for E: D – F = 2 Add F to both sides of the equation above: D – F + F = 2 + F which makes eq.5a) D = 2 + F
Hint #4
In eq.4, substitute 2×B for A, 2 + F for D (from eq.5a), and 2 for E: 2×B + F = B + 2 + F + 2 which becomes 2×B + F = B + F + 4 Subtract F and B from each side of the above equation: 2×B + F – F – B = B + F + 4 – F – B which simplifies to B = 4 making A = 2×B = 2 × 4 = 8
Hint #5
Substitute 8 for A, 4 for B, and 2 for C in eq.3: C + D = 8 + 4 + 2 which becomes C + D = 14 Subtract D from each side of the above equation: C + D – D = 14 – D which makes eq.3a) C = 14 – D
Hint #6
Substitute 4 for B, and 14 – D for C (from eq.3a) in eq.1: D = 4 + 14 – D which becomes D = 18 – D Add D to both sides of the above equation: D + D = 18 – D + D which becomes 2×D = 18 Divide both sides by 2: 2×D ÷ 2 = 18 ÷ 2 which makes D = 9 making C = 14 – 9 = 5 (from eq.3a)
Solution
Substitute 9 for D in eq.5a: 9 = 2 + F Subtract 2 from each side of the above equation: 9 – 2 = 2 + F – 2 which makes 7 = F and ABCDEF = 845927