Puzzle for July 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 non-negative integer.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add D and F to both sides of eq.4: C – D – F + D + F = A + D + F which becomes C = A + D + F Subtract A from both sides of the above equation: C – A = A + D + F – A which becomes eq.4a) C – A = D + F
Hint #2
In eq.3, replace C – A with D + F (from eq.4a): B + D = D + F Subtract D from both sides: B + D – D = D + F – D which makes B = F
Hint #3
In eq.2, replace F with B: E + B = B Subtract B from both sides of the equation above: E + B – B = B – B which makes E = 0
Hint #4
In eq.5, substitute B for F, and 0 for E: D + B = B + 0 Subtract B from each side of the equation above: D + B – B = B + 0 – B which makes D = 0
Hint #5
Substitute 0 for D in eq.3: B + 0 = C – A Add A to each side of the above equation: B + 0 + A = C – A + A which becomes eq.3a) B + A = C
Hint #6
eq.6 may be written as: 10×A + B – C = A + C Substitute (B + A) for C (from eq.3a) in the above equation: 10×A + B – (B + A) = A + (B + A) which becomes 10×A + B – B – A = A + B + A which becomes 9×A = 2×A + B Subtract 2×A from both sides of the above equation: 9×A – 2×A = 2×A + B – 2×A which makes 7×A = B and also makes F = B = 7×A
Hint #7
Substitute 7×A for B in eq.3a: A + 7×A = C which makes 8×A = C
Solution
Substitute 7×A for B and F, 8×A for C, and 0 for D and E in eq.1: A + 7×A + 8×A + 0 + 0 + 7×A = 23 which simplifies to 23×A = 23 Divide both sides of the above equation by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = F = 7×A = 7 × 1 = 7 C = 8×A = 8 × 1 = 8 and ABCDEF = 178007