Puzzle for March 8, 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.6, replace E with C + F (from eq.4): B + F = C + C + F which becomes B + F = 2×C + F Subtract F from both sides of the above equation: B + F – F = 2×C + F – F which makes B = 2×C
Hint #2
In eq.2, replace B with 2×C: D = 2×C + C which makes D = 3×C
Hint #3
In eq.3, substitute 3×C for D: C + 3×C = A which makes 4×C = A
Hint #4
Substitute 4×C for A, and 3×C for D in eq.5: 4×C + 3×C = E which makes 7×C = E
Hint #5
Substitute 7×C for E in eq.4: 7×C = C + F Subtract C from each side of the equation above: 7×C – C = C + F – C which makes 6×C = F
Solution
Substitute 4×C for A, 2×C for B, 3×C for D, 7×C for E, and 6×C for F in eq.1: 4×C + 2×C + C + 3×C + 7×C + 6×C = 23 which simplifies to 23×C = 23 Divide both sides of the equation above by 23: 23×C ÷ 23 = 23 ÷ 23 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 2×C = 2 × 1 = 2 D = 3×C = 3 × 1 = 3 E = 7×C = 7 × 1 = 7 F = 6×C = 6 × 1 = 6 and ABCDEF = 421376