Puzzle for June 24, 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.3, replace C with B + E (from eq.2): B + B + E = A + E which becomes 2×B + E = A + E Subtract E from both sides of the above equation: 2×B + E - E = A + E - E which makes eq.3a) 2×B = A
Hint #2
In eq.5, replace A with E - D (from eq.4): E - D + B = D + E In the above equation, add D to both sides, and subtract E from both sides: E - D + B + D - E = D + E + D - E which simplifies to B = 2×D
Hint #3
In eq.3a, substitute (2×D) for B: 2×(2×D) = A which makes 4×D = A
Hint #4
Substitute 4×D for A in eq.4: E - D = 4×D Add D to each side of the above equation: E - D + D = 4×D + D which makes E = 5×D
Hint #5
Substitute 4×D for A in eq.6: D = 4×D - F Add F to both sides, and subtract D from both sides: D + F - D = 4×D - F + F - D which makes F = 3×D
Hint #6
Substitute 2×D for B, and 5×D for E in eq.2: C = 2×D + 5×D which makes C = 7×D
Solution
Substitute 4×D for A, 2×D for B, 7×D for C, 5×D for E, and 3×D for F in eq.1: 4×D + 2×D + 7×D + D + 5×D + 3×D = 22 which simplifies to 22×D = 22 Divide both sides of the above equation by 22: 22×D ÷ 22 = 22 ÷ 22 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 2×D = 2 × 1 = 2 C = 7×D = 7 × 1 = 7 E = 5×D = 5 × 1 = 5 F = 3×D = 3 × 1 = 3 and ABCDEF = 427153