Puzzle for January 29, 2021 ( )
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.
* "A mod D" equals the remainder of (A ÷ D).
Scratchpad
Help Area
Hint #1
eq.5 may be written as: C – D = (A + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × (C – D) = 3 × (A + C + E) ÷ 3 which becomes 3×C – 3×D = A + C + E which is the same as eq.5a) 3×C – 3×D = A + E + C
Hint #2
In eq.5a, replace A + E with B + D (from eq.2): 3×C – 3×D = B + D + C Subtract D and C from each side of the above equation: 3×C – 3×D – D – C = B + D + C – D – C which becomes eq.5b) 2×C – 4×D = B
Hint #3
Add C to both sides of eq.1: A – C + C = C – B + C which becomes A = 2×C – B In the equation above, substitute (2×C – 4×D) for B (from eq.5b): A = 2×C – (2×C – 4×D) which becomes A = 2×C – 2×C + 4×D which makes A = 4×D
Hint #4
Substitute 4×D for A in eq.6: E = 4×D mod D which may be written as E = remainder of (4×D ÷ D) which means 0 = remainder of (4×D ÷ D) and therefore makes 0 = E
Hint #5
Substitute 4×D for A, and 0 for E into eq.2: B + D = 4×D + 0 Subtract D from each side of the equation above: B + D – D = 4×D + 0 – D which makes B = 3×D
Hint #6
Substitute 3×D for B in eq.5b: 2×C – 4×D = 3×D Add 4×D to both sides of the equation above: 2×C – 4×D + 4×D = 3×D + 4×D which becomes 2×C = 7×D Divide both sides by 2: 2×C ÷ 2 = 7×D ÷ 2 which makes C = 3½×D
Hint #7
In eq.4, substitute 3½×D for C, and 3×D for B: D = (3½×D ÷ D) – (F ÷ 3×D) which becomes D = 3½ – (F ÷ 3×D) In the above equation, add (F ÷ 3×D) to both sides, and subtract D from both sides: D + (F ÷ 3×D) – D = 3½ – (F ÷ 3×D) + (F ÷ 3×D) – D which becomes F ÷ 3×D = 3½ – D Multiply both sides by 3×D: F ÷ 3×D × 3×D = (3½ – D) × 3×D which becomes eq.4a) F = 10½×D – 3×D²
Solution
Substitute 3½×D for C, 0 for E, 10½×D – 3×D² for F (from eq.4a), and 4×D for A in eq.3: 3½×D – 0 + 10½×D – 3×D² = 4×D × D which becomes 14×D – 3×D² = 4×D² Add 3×D² to both sides of the above equation: 14×D – 3×D² + 3×D² = 4×D² + 3×D² which makes 14×D = 7×D² Since D ≠ 0 (from eq.4), divide both sides by 7×D: 14×D ÷ 7×D = 7×D² ÷ 7×D which makes 2 = D making A = 4×D = 4×2 = 8 B = 3×D = 3×2 = 6 C = 3½×D = 3½×2 = 7 F = 10½×D – 3×D² = 10½×2 – 3×2² = 21 – 3×4 = 21 – 12 = 9 (from eq.4a) and ABCDEF = 867209