Puzzle for April 9, 2023 ( )
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
In eq.2, replace A with C + D + F (from eq.1): C + D + F + D = B + C + E + F which becomes C + 2×D + F = B + C + E + F Subtract C and F from both sides of the above equation: C + 2×D + F - C - F = B + C + E + F - C - F which becomes eq.2a) 2×D = B + E
Hint #2
Multiply both sides of eq.3 by E: E × F = E × B ÷ E which becomes eq.3a) E × F = B
Hint #3
In eq.4, replace B with E × F (from eq.3a): E × F = D ÷ F Multiply both sides of the above equation by F: F × E × F = F × D ÷ F which becomes eq.4a) F² × E = D
Hint #4
In eq.5, substitute (F² × E) for D (from eq.4a), and E × F for B (from eq.3a): (F² × E) + E = E × F × C Since E ≠ 0 (from eq.3), divide both sides of the above equation by E: ((F² × E) + E) ÷ E = (E × F × C) ÷ E which becomes eq.5a) F² + 1 = F × C
Hint #5
Subtract F² from each side of eq.5a: F² + 1 - F² = (F × C) - F² which becomes 1 = (F × C) - F² which is the same as 1 = F × (C - F) Since F ≠ 0 (from eq.4), divide both sides of the above equation by F: 1 ÷ F = F × (C - F) ÷ F which becomes eq.5b) 1 ÷ F = C - F
Hint #6
Since C and F are integers, then in eq.5b: C - F = an integer which makes 1 ÷ F = an integer If F ≥ 2, then: 1 ÷ F = a fraction So therefore: F = 1
Hint #7
Substitute 1 for F in eq.5a: 1² + 1 = 1 × C which becomes 1 + 1 = C which makes 2 = C
Hint #8
Substitute 1 for F in eq.4a: 1² × E = D which becomes 1 × E = D which makes E = D
Hint #9
Substitute 1 for F in eq.3a: E × 1 = B which makes E = B and makes D = E = B
Hint #10
Substitute 2 for C, and 1 for F in eq.1: A = 2 + D + 1 which makes eq.1a) A = 3 + D
Hint #11
Substitute D for B and E, (3 + D) for A (from eq.1a), and 2 for C in eq.6: D + D + D = (3 + D) × 2 which becomes 3×D = 6 + 2×D Subtract 2×D from each side of the equation above: 3×D - 2×D = 6 + 2×D - 2×D which makes D = 6 and also makes D = E = B = 6
Solution
Substitute 6 for D in eq.1a: A = 3 + 6 which makes A = 9 and makes ABCDEF = 962661