Puzzle for February 4, 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 with B + E (from eq.5): C + E = B + E + B Subtract E from both sides of the above equation: C + E - E = B + E + B - E which makes C = 2×B
Hint #2
In eq.4, replace C with 2×B: E = B + 2×B which makes E = 3×B
Hint #3
In eq.5, substitute 3×B for E: B + 3×B = A which means 4×B = A
Hint #4
Substitute 4×B for A, and 3×B for E in eq.3: B + 3×B + F = 4×B + D which becomes 4×B + F = 4×B + D Subtract 4×B from each side of the above equation: 4×B + F - 4×B = 4×B + D - 4×B which makes F = D
Hint #5
Substitute D for F, 2×B for C, and 3×B for E in eq.6: D + D = 3×B + 2×B which becomes 2×D = 5×B Divide both sides of the above equation by 2: 2×D ÷ 2 = 5×B ÷ 2 which makes D = 2½×B
Solution
Substitute 4×B for A, 2×B for C, 2½×B for D and F, and 3×B for E in eq.1: 4×B + B + 2×B + 2½×B + 3×B + 2½×B = 30 which simplifies to 15×B = 30 Divide both sides by 15: 15×B ÷ 15 = 30 ÷ 15 which makes B = 2 making A = 4×B = 4 × 2 = 8 C = 2×B = 2 × 2 = 4 D = F = 2½×B = 2½ × 2 = 5 E = 3×B = 3 × 2 = 6 and ABCDEF = 824565