Puzzle for September 11, 2020 ( )
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.
* "C mod D" equals the remainder of (C ÷ D).
Scratchpad
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Hint #1
Add the left and right sides of eq.5 to the left and right sides of eq.4, respectively. C + E + C + F = A + D + B + D which becomes 2×C + E + F = A + B + 2×D Subtract 2×C from each side of the equation above: 2×C + E + F – 2×C = A + B + 2×D – 2×C which becomes eq.4a) E + F = A + B + 2×D – 2×C
Hint #2
In eq.2, replace E + F with A + B + 2×D – 2×C (from eq.4a): D + A + B + 2×D – 2×C = A + B which becomes A + B + 3×D – 2×C = A + B In the above equation, subtract A and B from both sides, and add 2×C to both sides: A + B + 3×D – 2×C – A – B + 2×C = A + B – A – B + 2×C which becomes 3×D = 2×C Divide both sides by 2: 3×D ÷ 2 = 2×C ÷ 2 which makes 1½×D = C
Hint #3
In eq.6, replace C with 1½×D: A = 1½×D mod D which means A = remainder of 1½×D ÷ D which makes A = ½×D
Hint #4
In eq.4, substitute 1½×D for C, and ½×D for A: 1½×D + E = ½×D + D which becomes 1½×D + E = 1½×D Subtract 1½×D from each side of the above equation: 1½×D + E – 1½×D = 1½×D – 1½×D which means E = 0
Hint #5
Substitute 1½×D for C in eq.5: 1½×D + F = B + D Subtract D from each side of the above equation: 1½×D + F – D = B + D – D which becomes eq.5a) ½×D + F = B
Hint #6
Substitute ½×D for A, and 1½×D for C in eq.3: B + F = ½×D + 1½×D + D which becomes eq.3a) B + F = 3×D
Hint #7
Substitute ½×D + F for B (from eq.5a) into eq.3a: ½×D + F + F = 3×D which becomes ½×D + 2×F = 3×D Subtract ½×D from each side of the equation above: ½×D + 2×F – ½×D = 3×D – ½×D which makes 2×F = 2½×D Divide both sides by 2: 2×F ÷ 2 = 2½×D ÷ 2 which makes F = 1¼×D
Hint #8
Substitute 1¼×D for F in eq.5a: ½×D + 1¼×D = B which makes 1¾×D = B
Solution
Substitute ½×D for A, 1¾×D for B, 1½×D for C, 0 for E, and 1¼×D for F in eq.1: ½×D + 1¾×D + 1½×D + D + 0 + 1¼×D = 24 which simplifies to 6×D = 24 Divide both sides of the equation above by 6: 6×D ÷ 6 = 24 ÷ 6 which means D = 4 making A = ½×D = ½ × 4 = 2 B = 1¾×D = 1¾ × 4 = 7 C = 1½×D = 1½ × 4 = 6 F = 1¼×D = 1¼ × 4 = 5 and ABCDEF = 276405