Puzzle for November 5, 2022 ( )
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 is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add B and D to both sides of eq.4: D - B + B + D = C - D + F + B + D which becomes eq.4a) 2×D = C + F + B Add D and A to both sides of eq.2: E - D + D + A = D - A + D + A which becomes eq.2a) E + A = 2×D
Hint #2
In eq.4a, replace 2×D with E + A (from eq.2a): E + A = C + F + B which may be written as eq.4b) A + E = B + C + F
Hint #3
In eq.3, replace A + E with B + C + F (from eq.4b): B + C = B + C + F + F which becomes B + C = B + C + 2×F Subtract B and C from both sides of the above equation: B + C - B - C = B + C + 2×F - B - C which makes 0 = 2×F which means 0 = F
Hint #4
In eq.4a, substitute 0 for F: 2×D = C + 0 + B which becomes eq.4c) 2×D = C + B
Hint #5
eq.1 may be written as: E + A + C + B + D + F = 25 In the equation above, substitute 2×D for E + A (from eq.2a) and for C + B (from eq.4c), and 0 for F: 2×D + 2×D + D + 0 = 25 which becomes 5×D = 25 Divide both sides by 5: 5×D ÷ 5 = 25 ÷ 5 which makes D = 5
Hint #6
Substitute 5 for D in eq.2a: E + A = 2×5 which becomes eq.2b) E + A = 10
Hint #7
Substitute 5 for D in eq.4c: 2×5 = C + B which becomes eq.4d) 10 = C + B
Hint #8
eq.6 may be written as: B × D = E + A + C + B Substitute 5 for D, and 10 for E + A (from eq.2b) and for C + B (from eq.4d) in the above equation: B × 5 = 10 + 10 which makes B × 5 = 20 Divide both sides by 5: B × 5 ÷ 5 = 20 ÷ 5 which makes B = 4
Hint #9
Substitute 4 for B in eq.4d: 10 = C + 4 Subtract 4 from both sides of the above equation: 10 - 4 = C + 4 - 4 which makes 6 = C
Hint #10
eq.5 may be written as: eq.5a) 10×A + B - F = D + E + F Subtract A from each side of eq.2b: E + A - A = 10 - A which becomes eq.2c) E = 10 - A
Hint #11
In eq.5a, substitute 4 for B, 0 for F, 5 for D, and 10 - A for E (from eq.2c): 10×A + 4 - 0 = 5 + 10 - A + 0 which becomes 10×A + 4 = 15 - A In the above equation, subtract 4 from both sides, and add A to both sides: 10×A + 4 - 4 + A = 15 - A - 4 + A which makes 11×A = 11 Divide both sides by 11: 11×A ÷ 11 = 11 ÷ 11 which makes A = 1
Solution
Substitute 1 for A in eq.2c: E = 10 - 1 which makes E = 9 and makes ABCDEF = 146590