Puzzle for February 25, 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.
* "B mod D" equals the remainder of B ÷ D.
Scratchpad
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Hint #1
In eq.2, replace C with B + D (from eq.3): B + E = A + B + D Subtract B from both sides of the equation above: B + E – B = A + B + D – B which becomes eq.2a) E = A + D
Hint #2
In eq.1, replace A + D with E (from eq.2a): E = F
Hint #3
eq.6 may be written as: F = (B + C + D + E) ÷ 4 Multiply both sides of the equation above by 4: 4 × F = 4 × (B + C + D + E) ÷ 4 which becomes eq.6a) 4×F = B + C + D + E
Hint #4
In eq.4, substitute E for F: D + E + E = B + C – D which becomes D + 2×E = B + C – D Add D to both sides of the above equation: D + 2×E + D = B + C – D + D which becomes eq.4a) 2×D + 2×E = B + C
Hint #5
Substitute E for F, and 2×D + 2×E for B + C (from eq.4a) in eq.6a: 4×E = 2×D + 2×E + D + E which becomes 4×E = 3×D + 3×E Subtract 3×E from each side of the equation above: 4×E – 3×E = 3×D + 3×E – 3×E which makes E = 3×D and also makes F = E = 3×D
Hint #6
Substitute 3×D for F in eq.1: A + D = 3×D Subtract D from each side of the above equation: A + D – D = 3×D – D which makes A = 2×D
Hint #7
Substitute (3×D) for E, and B + D for C (from eq.3) in eq.4a: 2×D + 2×(3×D) = B + B + D which becomes 2×D + 6×D = 2×B + D which becomes 8×D = 2×B + D Subtract D from both sides of the above equation: 8×D – D = 2×B + D – D which makes 7×D = 2×B Divide both sides by 2: 7×D ÷ 2 = 2×B ÷ 2 which makes 3½×D = B
Hint #8
Substitute 3½×D for B in eq.3: C = 3½×D + D which makes C = 4½×D
Solution
Substitute E for F, and 3½×D for B in eq.5: E ÷ E = 3½×D mod D which means 1 = remainder of (3½×D ÷ D) which means 1 = ½×D Multiply both sides of the above equation by 2: 2 × 1 = 2 × ½×D which means 2 = D making A = 2×D = 2 × 2 = 4 B = 3½×D = 3½ × 2 = 7 C = 4½×D = 4½ × 2 = 9 E = F = 3×D = 3 × 2 = 6 and ABCDEF = 479266