Puzzle for June 29, 2019 ( )
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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
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Hint #1
Add D to each side of eq.4: D - A + D = B - D + D which becomes eq.4a) 2×D - A = B
Hint #2
In eq.2, replace B with 2×D - A (from eq.4a): F = 2×D - A + D which becomes eq.2a) F = 3×D - A
Hint #3
In eq.6, replace B with 2×D - A (from eq.4a): 2×D - A = A + C + D Subtract both A and D from each side of the above equation: 2×D - A - A - D = A + C + D - A - D which becomes eq.6a) D - 2×A = C
Hint #4
In eq.3, substitute D - 2×A for C (from eq.6a), 2×D - A for B (from eq.4a), and 3×D - A for F (from eq.2a): E - A + D - 2×A = 2×D - A + 3×D - A which becomes E - 3×A + D = 5×D - 2×A Add (3×A - D) to both sides of the above equation: E - 3×A + D + (3×A - D) = 5×D - 2×A + (3×A - D) which becomes eq.3a) E = 4×D + A
Hint #5
eq.5 may be written as: A + 10×D + E = 10×B + C In the above equation, substitute 4×D + A for E (from eq.3a), (2×D - A) for B (from eq.4a), and (D - 2×A) for C (from eq.6a): A + 10×D + 4×D + A = 10×(2×D - A) + (D - 2×A) which becomes 2×A + 14×D = 20×D - 10×A + D - 2×A which becomes 2×A + 14×D = 21×D - 12×A Add (12×A - 14×D) to both sides: 2×A + 14×D + (12×A - 14×D) = 21×D - 12×A + (12×A - 14×D) which simplifies to 14×A = 7×D Divide by both sides of the above equation by 7: 14×A ÷ 7 = 7×D ÷ 7 which means 2×A = D
Hint #6
Substitute (2×A) for D in eq.4a: 2×(2×A) - A = B which is the same as 4×A - A = B which makes 3×A = B
Hint #7
Substitute (2×A) for D in eq.2a: F = 3×(2×A) - A which is equivalent to F = 6×A - A which makes F = 5×A
Hint #8
Substitute 2×A for D in eq.6a: 2×A - 2×A = C which means 0 = C
Hint #9
Substitute (2×A) for D in eq.3a: E = 4×(2×A) + A which becomes E = 8×A + A which makes E = 9×A
Solution
Substitute 3×A for B, 0 for C, 2×A for D, 9×A for E, and 5×A for F in eq.1: A + 3×A + 0 + 2×A + 9×A + 5×A = 20 which simplifies to 20×A = 20 Divide by both sides 20: 20×A ÷ 20 = 20 ÷ 20 which means A = 1 making B = 3×A = 3 × 1 = 3 D = 2×A = 2 × 1 = 2 E = 9×A = 9 × 1 = 9 F = 5×A = 5 × 1 = 5 and ABCDEF = 130295