Puzzle for January 5, 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.
Scratchpad
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Hint #1
In eq.4, replace D with A + B (from eq.2): A + A + B = B + F which becomes 2×A + B = B + F Subtract B from both sides of the above equation: 2×A + B – B = B + F – B which makes 2×A = F
Hint #2
In eq.6, replace F with 2×A: C + 2×A – A = A + E which becomes C + A = A + E Subtract A from each side of the equation above: C + A – A = A + E – A which makes C = E
Hint #3
In eq.3, substitute 2×A for F: E = A + B + 2×A which becomes eq.3a) E = B + 3×A
Hint #4
In eq.5, substitute A + B for D (from eq.2), 2×A for F, and B + 3×A for E (from eq.3a): B + A + B – 2×A = B + 3×A + 2×A which becomes 2×B – A = B + 5×A In the equation above, add A to both sides, and subtract B from both sides: 2×B – A + A – B = B + 5×A + A – B which makes B = 6×A
Hint #5
Substitute 6×A for B in eq.3a: E = 6×A + 3×A which makes E = 9×A and also makes C = E = 9×A
Hint #6
Substitute 6×A for B in eq.2: D = A + 6×A which makes D = 7×A
Solution
Substitute 6×A for B, 9×A for C and E, 7×A for D, and 2×A for F in eq.1: A + 6×A + 9×A + 7×A + 9×A + 2×A = 34 which simplifies to 34×A = 34 Divide both sides of the above equation by 34: 34×A ÷ 34 = 34 ÷ 34 which means A = 1 making B = 6×A = 6 × 1 = 6 C = E = 9×A = 9 × 1 = 9 D = 7×A = 7 × 1 = 7 F = 2×A = 2 × 1 = 2 and ABCDEF = 169792