Puzzle for August 9, 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.
* "D mod A" equals the remainder of D divided by A.
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Hint #1
Multiply each side of eq.3 by both C and B: F ÷ C × C × B = A ÷ B × C × B which becomes F × B = A × C which may be written as B × F = A × C In eq.4, replace B × F with A × C: C × D = (A × C) + (A × C) which becomes C × D = 2 × (A × C) Divide both sides of the above equation by C: C × D ÷ C = 2 × (A × C) ÷ C which makes D = 2×A
Hint #2
In eq.6, replace D with 2×A: 2×A mod A = C – F Since the remainder of 2×A ÷ A = 0, then: 2×A mod A = 0 which means 0 = C – F Subtract F from both sides of the above equation: 0 + F = C – F + F which makes F = C
Hint #3
In eq.3, substitute C for F: C ÷ C = A ÷ B which makes 1 = A ÷ B Multiply both sides of the above equation by B: B × 1 = B × A ÷ B which makes B = A
Hint #4
Substitute A for B in eq.1: E = A × A which may be written as E = A²
Hint #5
Substitute A for B, A² for E, and 2×A for D in eq.2: A + (A × A²) = C × 2×A which may be written as A + A³ = C × 2×A Divide both sides of the above equation by 2×A: (A + A³) ÷ 2×A = C × 2×A ÷ 2×A which becomes (A ÷ 2×A) + (A³ ÷ 2×A) = C which becomes (1 ÷ 2) + (A² ÷ 2) = C which means ½ + ½×A² = C and also means eq.2a) F = C = ½ + ½×A²
Solution
Substitute A for B, 2×A for D, A² for E, and C for F in eq.5: (A + 2×A) ÷ A² = C ÷ C which becomes 3×A ÷ A² = 1 which becomes 3 ÷ A = 1 Multiply both sides of the above equation by A: A × 3 ÷ A = A × 1 which means 3 = A making B = A = 3 C = F = ½ + ½×A² = ½ + ½×3² = ½ + ½×9 = ½ + 4½ = 5 (from eq.2a) D = 2×A = 2×3 = 6 E = A² = 3² = 9 and ABCDEF = 335695