Puzzle for May 25, 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, CD, and EF are 2-digit numbers (not A×B, C×D, or E×F).
Scratchpad
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Hint #1
eq.5 may be written as: B + E = C + A + D In the above equation, replace A + D with E + F (from eq.3): B + E = C + E + F Subtract E from both sides: B + E - E = C + E + F - E which becomes eq.5a) B = C + F
Hint #2
Add A to both sides of eq.4: E - A + A = A + C + A which becomes eq.4a) E = C + 2×A In eq.2, substitute C + F for B (from eq.5a), and (C + 2×A) for E (from eq.4a): C + F - (C + 2×A) = A which is equivalent to C + F - C - 2×A = A which becomes F - 2×A = A Add 2×A to both sides: F - 2×A + 2×A = A + 2×A which makes F = 3×A
Hint #3
Add E to both sides of eq.2: B - E + E = A + E which makes eq.2a) B = A + E
Hint #4
Subtract 2×A from both sides of eq.4a: E - 2×A = C + 2×A - 2×A which makes eq.4b) E - 2×A = C
Hint #5
Substitute 3×A for F in eq.3: A + D = E + 3×A Subtract A from both sides of the equation above: A + D - A = E + 3×A - A which makes eq.3a) D = E + 2×A
Hint #6
eq.6 may be written as: 10×A + B + 10×C + D = 10×E + F Substitute A + E for B (from eq.2a), (E - 2×A) for C (from eq.4b), E + 2×A for D (from eq.3a), and 3×A for F in the above equation: 10×A + A + E + 10×(E - 2×A) + E + 2×A = 10×E + 3×A which is equivalent to 10×A + A + E + 10×E - 20×A + E + 2×A = 10×E + 3×A which becomes 12×E - 7×A = 10×E + 3×A Add (7×A - 10×E) to both sides: 12×E - 7×A + (7×A - 10×E) = 10×E + 3×A + (7×A - 10×E) which simplifies to 2×E = 10×A Divide both sides by 2: 2×E ÷ 2 = 10×A ÷ 2 which makes E = 5×A
Hint #7
Substitute 5×A for E in eq.2a: B = A + 5×A which means B = 6×A
Hint #8
Substitute 5×A for E in eq.4b: 5×A - 2×A = C which makes 3×A = C
Hint #9
Substitute 5×A for E in eq.3a: D = 5×A + 2×A which makes D = 7×A
Solution
Substitute 6×A for B, 3×A for C and F, 7×A for D, and 5×A for E in eq.1: A + 6×A + 3×A + 7×A + 5×A + 3×A = 25 which becomes 25×A = 25 Divide both sides of the above equation by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = 6×A = 6 × 1 = 6 C = F = 3×A = 3 × 1 = 3 D = 7×A = 7 × 1 = 7 E = 5×A = 5 × 1 = 5 and ABCDEF = 163753