Puzzle for May 15, 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.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add both C and D to each side of eq.4: A - D + C + D = B - C + C + D which becomes A + C = B + D In eq.3, replace A + C with B + D: D + F = B + D Subtract D from each side: D + F - D = B + D - D which makes F = B
Hint #2
In eq.2, replace F with B: B = B + D Subtract B from both sides: B - B = B + D - B which means 0 = D
Hint #3
eq.6 may be written as: 10×C + D - F = 10×A + B In the equation above, substitute 0 for D, and B for F: 10×C + 0 - B = 10×A + B Add (B - 10×A) to each side: 10×C + 0 - B + (B - 10×A) = 10×A + B + (B - 10×A) which simplifies to 10×C - 10×A = 2×B Divide both sides by 2: (10×C - 10×A) ÷ 2 = 2×B ÷ 2 which becomes eq.6a) 5×C - 5×A = B
Hint #4
In eq.4, substitute 0 for D, and 5×C - 5×A for B (from eq.6a): A - 0 = 5×C - 5×A - C which becomes A = 4×C - 5×A Add 5×A to both sides of the above equation: A + 5×A = 4×C - 5×A + 5×A which becomes 6×A = 4×C Divide both sides by 4: 6×A ÷ 4 = 4×C ÷ 4 which makes 1½×A = C
Hint #5
Substitute (1½×A) for C in eq.6a: 5×(1½×A) - 5×A = B which is equivalent to 7½×A - 5×A = B which makes 2½×A = B and which also makes F = B = 2½×A
Hint #6
Substitute 2½×A for B, and 1½×A for C in eq.5: 2½×A + 1½×A = E which makes 4×A = E
Solution
Substitute 2½×A for B and F, 1½×A for C, 0 for D, and 4×A for E in eq.1: A + 2½×A + 1½×A + 0 + 4×A + 2½×A = 23 which simplifies to 11½×A = 23 Divide both sides of the equation above by 11½: 11½×A ÷ 11½ = 23 ÷ 11½ which means A = 2 making B = F = 2½×A = 2½ × 2 = 5 C = 1½×A = 1½ × 2 = 3 E = 4×A = 4 × 2 = 8 and ABCDEF = 253085