Puzzle for August 22, 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.
* "F mod D" equals the remainder of F ÷ D.
Scratchpad
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Hint #1
In eq.2, add E to both sides, and subtract C from both sides: D – E + E – C = C + E + E – C which becomes D – C = 2×E In eq.4, replace D – C with 2×E: 2×E = B ÷ E Multiply both sides of the equation above by E: 2×E × E = B ÷ E × E which makes eq.4a) 2×E² = B
Hint #2
In eq.3, replace C with B – D (from eq.1): B + D – F = B – D – D + F which becomes B + D – F = B – 2×D + F In the above equation, subtract B from each side, and add F and 2×D to each side: B + D – F – B + F + 2×D = B – 2×D + F – B + F + 2×D which simplifies to 3×D = 2×F Divide both sides by 2: 3×D ÷ 2 = 2×F ÷ 2 which makes 1½×D = F
Hint #3
In eq.6, substitute 1½×D for F: 1½×D mod D = D ÷ E which means remainder of (1½×D ÷ D) = D ÷ E which means eq.6a) ½×D = D ÷ E
Hint #4
Multiply both sides of eq.6a by both 2 and E: (½) × D × 2 × E = D ÷ E × 2 × E which becomes D × E = D × 2 Divide both sides of the above equation by D: D × E ÷ D = D × 2 ÷ D which makes E = 2
Hint #5
Substitute 2 for E in eq.4a: 2×2² = B which becomes 2×4 = B which makes 8 = B
Hint #6
Substitute 2 for E in eq.2: D – 2 = C + 2 Add 2 to both sides of the equation above: D – 2 + 2 = C + 2 + 2 which becomes eq.2a) D = C + 4
Hint #7
Substitute 8 for B, and (C + 4) for D (from eq.2a) in eq.1: C = 8 – (C + 4) which is equivalent to C = 8 – C – 4 which becomes C = 4 – C Add C to both sides of the above equation: C + C = 4 – C + C which makes 2×C = 4 Divide both sides by 2: 2×C ÷ 2 = 4 ÷ 2 which makes C = 2
Hint #8
Substitute 8 for B, 2 for C, and 2 for E in eq.5: A – 8 = 2 ÷ 2 which becomes A – 8 = 1 Add 8 to both sides of the equation above: A – 8 + 8 = 1 + 8 which makes A = 9
Solution
Substitute 2 for C in eq.2a: D = 2 + 4 which makes D = 6 making F = 1½×D = 1½ × 6 = 9 and ABCDEF = 982629