Puzzle for December 29, 2020 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
In eq.1, replace B with D + E (from eq.3), and replace A with C + F (from eq.2): D + E – D = C + F – C which makes E = F
Hint #2
eq.6 may be written as: A + E + F = 10×E + F Subtract E and F from both sides of the above equation: A + E + F – E – F = 10×E + F – E – F which makes A = 9×E
Hint #3
In eq.5, replace F with E, and A with 9×E: E = 9×E ÷ B Multiply both sides of the above equation by B: E × B = 9×E ÷ B × B which becomes E × B = 9×E Divide both sides by E: E × B ÷ E = 9×E ÷ E which makes B = 9
Hint #4
In eq.2, substitute E for F, and 9×E for A: C + E = 9×E Subtract E from both sides of the equation above: C + E – E = 9×E – E which makes C = 8×E
Hint #5
Substitute 8×E for C in eq.4: E = 8×E ÷ D Multiply both sides of the above equation by D: E × D = 8×E ÷ D × D which becomes E × D = 8×E Divide both sides by E: E × D ÷ E = 8×E ÷ E which makes D = 8
Solution
Substitute 8 for D, and 9 for B in eq.3: 8 + E = 9 Subtract 8 from each side of the above equation: 8 + E – 8 = 9 – 8 which means E = 1 making A = 9×E = 9 × 1 = 9 C = 8×E = 8 × 1 = 8 F = E = 1 and ABCDEF = 998811