Puzzle for March 23, 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.
Scratchpad
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Hint #1
In eq.6, replace B with A + E (from eq.2): A – F = A + E – A which becomes A – F = E Add F to both sides of the above equation: A – F + F = E + F which becomes eq.6a) A = E + F
Hint #2
In eq.4, replace A with E + F (from eq.6a), and replace C with D + F (from eq.3): E + F = D + F + D which becomes E + F = 2×D + F Subtract F from each side of the equation above: E + F – F = 2×D + F – F which makes E = 2×D
Hint #3
In eq.5, substitute A + E for B (from eq.2): A + E + E = C + F which becomes eq.5a) A + 2×E = C + F
Hint #4
Substitute C + D for A (from eq.4), and (2×D) for E in eq.5a: C + D + 2×(2×D) = C + F which becomes C + D + 4×D = C + F which becomes C + 5×D = C + F Subtract C from both sides of the above equation: C + 5×D – C = C + F – C which makes 5×D = F
Hint #5
Substitute 5×D for F in eq.3: C = D + 5×D which makes C = 6×D
Hint #6
Substitute 2×D for E, and 5×D for F in eq.6a: A = 2×D + 5×D which makes A = 7×D
Hint #7
Substitute 7×D for A, and 2×D for E in eq.2: B = 7×D + 2×D which makes B = 9×D
Solution
Substitute 7×D for A, 9×D for B, 6×D for C, 2×D for E, and 5×D for F in eq.1: 7×D + 9×D + 6×D + D + 2×D + 5×D = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 9×D = 9 × 1 = 9 C = 6×D = 6 × 1 = 6 E = 2×D = 2 × 1 = 2 F = 5×D = 5 × 1 = 5 and ABCDEF = 796125