Puzzle for June 5, 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.
* "F mod D" equals the remainder of (F ÷ D).
Scratchpad
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Hint #1
In eq.4, add A and F to both sides, and subtract D from both sides: D – F + A + F – D = F – A – D + A + F – D which simplifies to eq.4a) A = 2×F – 2×D In eq.3, add A and F to both sides, and subtract D from both sides: F – D + A + F – D = C + D – A – F + A + F – D which simplifies to eq.3a) 2×F – 2×D + A = C
Hint #2
In eq.3a, replace 2×F – 2×D with A (from eq.4a): A + A = C which means 2×A = C
Hint #3
In eq.2, replace C with 2×A: A + 2×A = D + F which becomes eq.2a) 3×A = D + F
Hint #4
In eq.2a, substitute (2×F – 2×D) for A (from eq.4a): 3×(2×F – 2×D) = D + F which becomes 6×F – 6×D = D + F In the above equation, add 6×D to both sides, and subtract F from both sides: 6×F – 6×D + 6×D – F = D + F + 6×D – F which becomes 5×F = 7×D Divide both sides by 5: 5×F ÷ 5 = 7×D ÷ 5 which makes F = 1⅖×D
Hint #5
Substitute 2×A for C, and (1⅖×D) for F in eq.5: 2×A ÷ A = (1⅖×D) mod D which becomes eq.5a) 2 = (1⅖×D) mod D A property of the mod operation is -- if both numbers (or terms) have a factor in common, it may be factored out. Mathematically, the above property is expressed as: (m×D) mod (n×D) = (m mod n) × D
Hint #6
Applying the above property to eq.5a yields: 2 = (1⅖×D) mod (1×D) which becomes 2 = (1⅖ mod 1) × D which means 2 = remainder (1⅖ ÷ 1) × D which makes 2 = ⅖ × D Divide both sides of the equation above by ⅖: 2 ÷ ⅖ = ⅖×D ÷ ⅖ which makes 5 = D making F = 1⅖×D = 1⅖ × 5 = 7
Hint #7
Substitute 7 for F, and 5 for D in eq.4a: A = 2×7 – 2×5 which becomes A = 14 – 10 which makes A = 4 making C = 2×A = 2 × 4 = 8
Solution
Substitute 4 for A, 8 for C, 5 for D, and 7 for F in eq.1: 4 + B + 8 + 5 + E + 7 = 24 which becomes 24 + B + E = 24 Subtract 24 from each side of the above equation: 24 + B + E – 24 = 24 – 24 which makes B + E = 0 Since B and E are non-negative integers, the above equation makes: B = 0 and E = 0 making ABCDEF = 408507