Puzzle for July 27, 2022 ( )
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, replace B with A + F (from eq.2), and E with A + D (from eq.3): A + F = D + A + D which becomes A + F = 2×D + A Subtract A from each side of the above equation: A + F – A = 2×D + A – A which makes F = 2×D
Hint #2
In eq.2, replace F with 2×D: eq.2a) B = A + 2×D
Hint #3
In eq.6, substitute A + 2×D for B (from eq.2a), and 2×D for F: A – D + E = A + 2×D + D + 2×D which becomes A – D + E = A + 5×D In the equation above, subtract A from both sides, and add D to both sides: A – D + E – A + D = A + 5×D – A + D which makes E = 6×D
Hint #4
Substitute 6×D for E in eq.3: 6×D = A + D Subtract D from each side of the above equation: 6×D – D = A + D – D which makes 5×D = A
Hint #5
Substitute 5×D for A in eq.2a: B = 5×D + 2×D which makes B = 7×D
Hint #6
Substitute 6×D for E, 5×D for A, 7×D for B, and 2×D for F in eq.5: C + 6×D = 5×D + 7×D + 2×D which becomes C + 6×D = 14×D Subtract 6×D from both sides of the equation above: C + 6×D – 6×D = 14×D – 6×D which makes C = 8×D
Solution
Substitute 5×D for A, 7×D for B, 8×D for C, 6×D for E, and 2×D for F in eq.1: 5×D + 7×D + 8×D + D + 6×D + 2×D = 29 which simplifies to 29×D = 29 Divide both sides of the above equation by 29: 29×D ÷ 29 = 29 ÷ 29 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 7×D = 7 × 1 = 7 C = 8×D = 8 × 1 = 8 E = 6×D = 6 × 1 = 6 F = 2×D = 2 × 1 = 2 and ABCDEF = 578162