Puzzle for May 21, 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 DE are 2-digit numbers (not A×B or D×E).
Scratchpad
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Hint #1
Add both E and F to each side of eq.4: D + E + F = B + C - E - F + E + F which becomes D + E + F = B + C In eq.2, replace B + C with D + E + F: D + E + F = A + E + F Subtract both E and F from each side of the equation above: D + E + F - E - F = A + E + F - E - F which makes D = A
Hint #2
In eq.5, substitute A for D: E - A = A + A Add A to both sides of the equation above: E - A + A = A + A + A which makes E = 3×A
Hint #3
eq.6 may be written as: 10×A + B = 10×D + E + B - F Subtract B from both sides of the above equation: 10×A + B - B = 10×D + E + B - F - B which becomes 10×A = 10×D + E - F Substitute A for D, and 3×A for E: 10×A = 10×A + 3×A - F Add (F - 10×A) to both sides: 10×A + (F - 10×A) = 10×A + 3×A - F + (F - 10×A) which simplifies to F = 3×A
Hint #4
Substitute 3×A for E and F in eq.3: C + 3×A = B + 3×A Subtract 3×A from each side of the above equation: C + 3×A - 3×A = B + 3×A - 3×A which makes C = B
Hint #5
Substitute B for C, and 3×A for E and F in eq.2: B + B = A + 3×A + 3×A which becomes 2×B = 7×A Divide both sides by 2: 2×B ÷ 2 = 7×A ÷ 2 which makes B = 3½×A which also makes C = B = 3½×A
Solution
Substitute 3½×A for B and C, A for D, and 3×A for E and F in eq.1: A + 3½×A + 3½×A + A + 3×A + 3×A = 30 which simplifies to 15×A = 30 Divide both sides by 15: 15×A ÷ 15 = 30 ÷ 15 which means A = 2 making B = C = 3½×A = 3½ × 2 = 7 D = A = 2 E = F = 3×A = 3 × 2 = 6 and ABCDEF = 277266