Puzzle for June 18, 2021 ( )
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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 E" equals the remainder of (B ÷ E}.
We extend our thanks to Tom H for sending us this puzzle. Thank you, Tom!
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Hint #1
Multiply both sides of eq.1 by E and F: (A ÷ E) × (E × F) = (B ÷ F) × (E × F) which becomes eq.1a) A × F = B × E Since A, B, E, and F ≠ 0 (from eq.5, eq.2, and eq.1), divide the left and right sides of eq.4 by the left and right sides of eq.1a, respectively: (E × F) ÷ (A × F) = (A × B) ÷ (B × E) which becomes eq.1b) E ÷ A = A ÷ E
Hint #2
Multiply each side of eq.1b by both A and E: (E ÷ A) × (A × E) = (A ÷ E) × (A × E) which becomes E² = A² which means E = A
Hint #3
In eq.4, replace E with A: A × F = A × B Since A ≠ 0 (from eq.5), divide both sides of the above equation by A: (A × F) ÷ A = (A × B) ÷ A which makes F = B
Hint #4
In eq.3, replace F with B: C = B × B which may be written as eq.3a) C = B²
Hint #5
In eq.5, substitute B for F, and B² for C (from eq.3a): B = (B² – B) ÷ A which may be written as B = (B × (B – 1)) ÷ A Since B ≠ 0 (from eq.2), divide both sides of the above equation by B, and multiply both sides by A: B ÷ B × A = ((B × (B – 1)) ÷ A) ÷ B × A which becomes A = 1 × (B – 1) which makes A = B – 1 Add 1 to both sides: A + 1 = B – 1 + 1 which makes A + 1 = B and also makes eq.5a) F = B = A + 1
Hint #6
Substitute (A + 1) for B (from eq.5a) in eq.3a: eq.3b) C = (A + 1)²
Hint #7
Substitute A for E, (A + 1)² for C (from eq.3b), and (A + 1) for B and F (from eq.5a) in eq.2: A + A = ((A + 1)² + (A + 1)) ÷ (A + 1) which becomes 2×A = ((A + 1) + 1) which becomes 2×A = A + 2 Subtract A from each side of the above equation: 2×A – A = A + 2 – A which makes A = 2 and makes B = F = A + 1 = 2 + 1 = 3 (from eq.5a) C = (A + 1)² = (2 + 1)² = (3)² = 9 (from eq.3b) E = A = 2
Solution
Substitute 3 for B, and 2 for E in eq.6: D = 3 mod 2 which means D = remainder of (3 ÷ 2) which makes D = 1 and makes ABCDEF = 239123