Puzzle for April 5, 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.
* DE is a 2-digit number (not D×E). ABC is a 3-digit number (not A×B×C).
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Hint #1
Subtract C from each side of eq.2: D - C = A + B + C - C which becomes eq.2a) D - C = A + B Add (D - E) to each side of eq.4: E - C + (D - E) = F - D + (D - E) which becomes -C + D = F - E which may be expressed as eq.4a) D - C = F - E
Hint #2
In eq.4a, replace D - C with A + B (from eq.2a), and replace F - E with A (from eq.5): A + B = A Subtract A from each side of the above equation: A + B - A = A - A which means 0 = B
Hint #3
In eq.3, substitute (F - E) for A with (from eq.5): C - (F - E) = E + F which is the same as C - F + E = E + F Add (F - E) to both sides of the above equation: C - F + E + (F - E) = E + F + (F - E) which simplifies to C = 2×F
Hint #4
In eq.4, substitute 2×F for C: E - 2×F = F - D Add (D - E + 2×F) to both sides of the above equation: E - 2×F + (D - E + 2×F) = F - D + (D - E + 2×F) which simplifies to eq.4b) D = 3×F - E
Hint #5
eq.6 may be written as: 100×A + 10×B + C = 10×D + E + C + E + F Subtract C from each side of the above equation: 100×A + 10×B + C - C = 10×D + E + C + E + F - C which becomes 100×A + 10×B = 10×D + 2×E + F Substitute 0 for B, and (3×F - E) for D (from eq.4b): 100×A + 10×0 = 10×(3×F - E) + 2×E + F which is the same as 100×A = 30×F - 10×E + 2×E + F which becomes eq.6a) 100×A = 31×F - 8×E
Hint #6
In eq.6a, substitute (F - E) for A (from eq.5): 100×(F - E) = 31×F - 8×E which is the same as 100×F - 100×E = 31×F - 8×E Add (100×E - 31×F) to both sides of the equation above: 100×F - 100×E + (100×E - 31×F) = 31×F - 8×E + (100×E - 31×F) which simplifies to 69×F = 92×E Divide both sides by 92: 69×F ÷ 92 = 92×E ÷ 92 which makes ¾×F = E
Hint #7
Substitute ¾×F for E in eq.5: F - ¾×F = A which makes ¼×F = A
Hint #8
Substitute ¾×F for E in eq.4b: D = 3×F - ¾×F which means D = 2¼×F
Solution
Substitute ¼×F for A, 0 for B, 2×F for C, 2¼×F for D, and ¾×F for E in eq.1: ¼×F + 0 + 2×F + 2¼×F + ¾×F + F = 25 which simplifies to 6¼×F = 25 Divide both sides by 6¼: 6¼×F ÷ 6¼ = 25 ÷ 6¼ which means F = 4 making A = ¼×F = ¼ × 4 = 1 C = 2×F = 2 × 4 = 8 D = 2¼×F = 2¼ × 4 = 9 E = ¾×F = ¾ × 4 = 3 and ABCDEF = 108934