Puzzle for August 8, 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.3, replace B with D + F (from eq.2): C + F = D + F – C In the above equation, subtract F from both sides, and add C to both sides: C + F – F + C = D + F – C – F + C which makes 2×C = D
Hint #2
In eq.4, replace D with 2×C: 2×C = A + C Subtract C from each side of the equation above: 2×C – C = A + C – C which makes C = A
Hint #3
In eq.6, substitute C for A, and 2×C for D: F = C + 2×C which makes F = 3×C
Hint #4
Substitute 2×C for D, and 3×C for F in eq.2: B = 2×C + 3×C which makes B = 5×C
Hint #5
Substitute 5×C for B, and 3×C for F in eq.5: E = 5×C + 3×C which makes E = 8×C
Solution
Substitute C for A, 5×C for B, 2×C for D, 8×C for E, and 3×C for F in eq.1: C + 5×C + C + 2×C + 8×C + 3×C = 20 which simplifies to 20×C = 20 Divide both sides of the above equation by 20: 20×C ÷ 20 = 20 ÷ 20 which means C = 1 making A = C = 1 B = 5×C = 5 × 1 = 5 D = 2×C = 2 × 1 = 2 E = 8×C = 8 × 1 = 8 F = 3×C = 3 × 1 = 3 and ABCDEF = 151283