Puzzle for August 14, 2019 ( )
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
Add the left and right sides of eq.2 to the left and right sides of eq.5, respectively: C – E + E – A = A – D + A + D which becomes C – A = 2×A Add A to both sides of the above equation: C – A + A = 2×A + A which makes C = 3×A
Hint #2
In eq.6, replace F with A + E (from eq.4): D + E – A = A + C + A + E In the above equation, add A to both sides, and subtract E from both sides: D + E – A + A – E = A + C + A + E + A – E which simplifies to D = 3×A + C Replace C with 3×A: D = 3×A + 3×A which makes D = 6×A
Hint #3
In eq.3, replace C with 3×A, and replace D with 6×A: B = 3×A + 6×A which makes B = 9×A
Hint #4
In eq.2, substitute 6×A for D: E – A = A + 6×A Add A to both sides: E – A + A = A + 6×A + A which makes E = 8×A
Hint #5
Substitute 8×A for E in eq.4: F = A + 8×A which means F = 9×A
Solution
Substitute 9×A for B and F, 3×A for C, 6×A for D, and 8×A for E in eq.1: A + 9×A + 3×A + 6×A + 8×A + 9×A = 36 which simplifies to 36×A = 36 Divide both sides of the above equation by 36: 36×A ÷ 36 = 36 ÷ 36 A = 1 making B = F = 9×A = 9 × 1 = 9 C = 3×A = 3 × 1 = 3 D = 6×A = 6 × 1 = 6 E = 8×A = 8 × 1 = 8 and ABCDEF = 193689