Puzzle for June 26, 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 and EF are 2-digit numbers (not D×E or E×F).
Scratchpad
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Hint #1
In eq.4, replace C with E + F (from eq.2): B + E + F = D + E Subtract both E and F from each side of the equation above: B + E + F - E - F = D + E - E - F which becomes eq.4a) B = D - F In eq.4a, replace D - F with A (from eq.3): B = A
Hint #2
In eq.5, substitute (D - F) for B (from eq.4a): E - F = D - (D - F) which is equivalent to E - F = D - D + F which becomes E - F = F Add F to both sides of the above equation: E - F + F = F + F which makes E = 2×F
Hint #3
In eq.2, substitute 2×F for E: C = 2×F + F which makes C = 3×F
Hint #4
eq.6 may be written as: 10×D + E - (10×E + F) = A + C which is the same as 10×D + E - 10×E - F = A + C which becomes 10×D - 9×E - F = A + C Substitute (2×F) for E, and 3×F for C in the equation above: 10×D - 9×(2×F) - F = A + 3×F which becomes 10×D - 18×F - F = A + 3×F which becomes eq.4a) 10×D - 19×F = A + 3×F
Hint #5
Substitute D - F for A (from eq.3) in eq.4a: 10×D - 19×F = D - F + 3×F which becomes 10×D - 19×F = D + 2×F In the equation above, add 19×F to both sides, and subtract D from both sides: 10×D - 19×F + 19×F - D = D + 2×F + 19×F - D which simplifies to 9×D = 21×F Divide both sides by 9: 9×D ÷ 9 = 21×F ÷ 9 which makes D = 2⅓×F
Hint #6
Substitute 2⅓×F for D in eq.3: A = 2⅓×F - F which makes A = 1⅓×F which also makes B = A = 1⅓×F
Solution
Substitute 1⅓×F for A and B, 3×F for C, 2⅓×F for D, and 2×F for E in eq.1: 1⅓×F + 1⅓×F + 3×F + 2⅓×F + 2×F + F = 33 which simplifies to 11×F = 33 Divide both sides of the above equation by 11: 11×F ÷ 11 = 33 ÷ 11 which means F = 3 making A = B = 1⅓×F = 1⅓ × 3 = 4 C = 3×F = 3×3 = 9 D = 2⅓×F = 2⅓ × 3 = 7 E = 2×F = 2×3 = 6 and ABCDEF = 449763