Puzzle for May 20, 2019 ( )
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 A + B with C + F (from eq.5): E + F = C + F + C Subtract F from both sides of the above equation: E + F - F = C + F + C - F which makes E = 2×C
Hint #2
In eq.4, substitute 2×C for E: D = C + 2×C which makes D = 3×C
Hint #3
Substitute 3×C for D, and 2×C for E in eq.6: F = C + 3×C + 2×C which makes F = 6×C
Hint #4
Substitute 3×C for D, 6×C for F, and 2×C for E in eq.3: B + 3×C + 6×C = A + 2×C which becomes B + 9×C = A + 2×C Subtract 2×C from each side: B + 9×C - 2×C = A + 2×C - 2×C which becomes eq.3a) B + 7×C = A
Hint #5
Substitute 6×C for F in eq.5: C + 6×C = A + B which makes 7×C = A + B In the above equation, substitute B + 7×C for A (from eq.3a): 7×C = B + 7×C + B which becomes 7×C = 2×B + 7×C Subtract 7×C from both sides: 7×C - 7×C = 2×B + 7×C - 7×C which makes 0 = 2×B which means 0 = B
Hint #6
Substitute 0 for B in eq.3a: 0 + 7×C = A which means 7×C = A
Solution
Substitute 7×C for A, 0 for B, 3×C for D, 2×C for E, and 6×C for F in eq.1: 7×C + 0 + C + 3×C + 2×C + 6×C = 19 which simplifies to 19×C = 19 Divide both sides by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = 7×C = 7 × 1 = 7 D = 3×C = 3 × 1 = 3 E = 2×C = 2 × 1 = 2 F = 6×C = 6 × 1 = 6 and ABCDEF = 701326