Puzzle for October 16, 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.
* "C mod D" equals the remainder of C ÷ D.
Scratchpad
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Hint #1
In eq.1, replace A with E + F (from eq.3): B + E = E + F + C Subtract E from each side of the above equation: B + E – E = E + F + C – E which becomes eq.1a) B = F + C
Hint #2
In eq.2, replace B with F + C (from eq.1a): F – C – E = F + C – D Add C, E, and D to both sides of the above equation: F – C – E + C + E + D = F + C – D + C + E + D which becomes F + D = F + 2×C + E Subtract F from both sides: F + D – F = F + 2×C + E – F which becomes eq.2a) D = 2×C + E
Hint #3
In eq.4, substitute 2×C + E for D (from eq.2a): C + E = 2×C + E – E which becomes C + E = 2×C Subtract C from both sides of the above equation: C + E – C = 2×C – C which makes E = C
Hint #4
Substitute E for C in eq.1: B + E = A + E Subtract E from each side of the equation above: B + E – E = A + E – E which makes B = A
Hint #5
Substitute C for E in eq.2a: D = 2×C + C which makes D = 3×C
Hint #6
Substitute A for B, and 3×C for D in eq.5: A ÷ A = C mod 3×C which means eq.5a) 1 = remainder (C ÷ 3×C) (implies C ≠ 0)
Hint #7
In a division problem, if the dividend is smaller than the divisor, and both are positive, then: remainder of (dividend ÷ divisor) = dividend Since C > 0 (from eq.5a), then: C < 3×C eq.5a then becomes: C = remainder (C ÷ 3×C) which means C = 1 making E = C = 1 D = 3×C = 3 × 1 = 3
Hint #8
Substitute 3 for D, and 1 for C and E in eq.6: 3 = average (1, 1, F) which may be written as 3 = (1 + 1 + F) ÷ 3 which becomes 3 = (2 + F) ÷ 3 Multiply each side of the equation above by 3: 3 × 3 = 3 × (2 + F) ÷ 3 which becomes 9 = 2 + F Subtract 2 from each side: 9 – 2 = 2 + F – 2 which makes 7 = F
Solution
Substitute 1 for E, and 7 for F in eq.3: 1 + 7 = A which makes 8 = A and also makes B = A = 8 making ABCDEF = 881317