Puzzle for May 19, 2021 ( )
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).
Scratchpad
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Hint #1
In eq.2, replace D + E with C – D (from eq.5): C = C – D + F In the above equation, subtract C from both sides, and add D to both sides: C – C + D = C – D + F – C + D which simplifies to D = F
Hint #2
In eq.4, replace F with D: B = D – D which makes B = 0
Hint #3
In eq.3, replace F with D: D + D = A – D – E which becomes 2×D = A – D – E Add D and E to both sides of the equation above: 2×D + D + E = A – D – E + D + E which becomes eq.3a) 3×D + E = A
Hint #4
Add D to both sides of eq.5: C – D + D = D + E + D which becomes eq.5a) C = 2×D + E eq.6 may be written as: eq.6a) A + B + C = 10×D + E + F
Hint #5
In eq.6a, substitute 3×D + E for A (from eq.3a), 0 for B, 2×D + E for C (from eq.5a), and D for F: 3×D + E + 0 + 2×D + E = 10×D + E + D which becomes 5×D + 2×E = 11×D + E Subtract 5×D and E from both sides of the equation above: 5×D + 2×E – 5×D – E = 11×D + E – 5×D – E which makes E = 6×D
Hint #6
Substitute 6×D for E in eq.3a: 3×D + 6×D = A which makes 9×D = A
Hint #7
Substitute 6×D for E in eq.5a: C = 2×D + 6×D which makes C = 8×D
Solution
Substitute 9×D for A, 0 for B, 8×D for C, 6×D for E, and D for F in eq.1: 9×D + 0 + 8×D + D + 6×D + D = 25 which simplifies to 25×D = 25 Divide both sides of the above equation by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 9×D = 9 × 1 = 9 C = 8×D = 8 × 1 = 8 E = 6×D = 6 × 1 = 6 F = D = 1 and ABCDEF = 908161