Puzzle for July 18, 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.6, replace A + B with F (from eq.4): F + D = C + F Subtract F from both sides of the above equation: F + D - F = C + F - F which means D = C
Hint #2
In eq.3, replace A + B with F (from eq.4), and replace D with C: F = C + C which makes F = 2×C
Hint #3
In eq.5, substitute E for A - B (from eq.2), and substitute 2×C for F: E + E = C + 2×C which means 2×E = 3×C Divide both sides of the equation above by 2: 2×E ÷ 2 = 3×C ÷ 2 which makes E = 1½×C
Hint #4
Substitute 1½×C for E in eq.2: eq.2a) A - B = 1½×C Substitute 2×C for F in eq.4: eq.4a) A + B = 2×C
Hint #5
Add the left and right sides of eq.4a to the left and right sides of eq.2a, respectively: A - B + A + B = 1½×C + 2×C which makes 2×A = 3½×C Divide both sides of the equation above by 2: 2×A ÷ 2 = 3½×C ÷ 2 which means A = 1¾×C
Hint #6
Substitute 1¾×C for A in eq.4a: 1¾×C + B = 2×C Subtract 1¾×C from both sides: 1¾×C + B - 1¾×C = 2×C - 1¾×C which makes B = ¼×C
Solution
Substitute 1¾×C for A, ¼×C for B, C for D, 1½×C for E, and 2×C for F in eq.1: 1¾×C + ¼×C + C + C + 1½×C + 2×C = 30 which simplifies to 7½×C = 30 Divide both sides of the above equation by 7½: 7½×C ÷ 7½ = 30 ÷ 7½ which means C = 4 making A = 1¾×C = 1¾ × C = 7 B = ¼×C = ¼ × C = 1 D = C = 4 E = 1½×C = 1½ × C = 6 F = 2×C = 2 × C = 8 and ABCDEF = 714468