Puzzle for April 29, 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
Add C to both sides of eq.6: A - C + C = B + D + C which becomes A = B + D + C In eq.3, replace A with B + D + C: B + C = B + D + C Subtract both B and C from each side of the equation above: B + C - B - C = B + D + C - B - C which makes 0 = D
Hint #2
In eq.5, replace D with 0: C - 0 = B - C which becomes C = B - C Add C to each side of the equation above: C + C = B - C + C which makes 2×C = B
Hint #3
In eq.3, substitute 2×C for B: 2×C + C = A which makes 3×C = A
Hint #4
Substitute 0 for D in eq.2: E = A + 0 which makes E = A which also makes E = A = 3×C
Hint #5
Substitute 0 for D, and 3×C for both A and E in eq.4: 0 + F = 3×C + 3×C which makes F = 6×C
Solution
Substitute 3×C for A and E, 2×C for B, 0 for D, and 6×C for F in eq.1: 3×C + 2×C + C + 0 + 3×C + 6×C = 15 which simplifies to 15×C = 15 Divide both sides by 15: 15×C ÷ 15 = 15 ÷ 15 which means C = 1 making A = E = 3×C = 3 × 1 = 3 B = 2×C = 2 × 1 = 2 F = 6×C = 6 × 1 = 6 and ABCDEF = 321036