Puzzle for November 30, 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.
Scratchpad
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Hint #1
In eq.5, replace F with A + E (from eq.6): E + A + E = A + D which becomes A + 2×E = A + D Subtract A from both sides of the above equation: A + 2×E – A = A + D – A which makes 2×E = D
Hint #2
In eq.4, replace D with 2×E: 2×E = C – E Add E to both sides of the equation above: 2×E + E = C – E + E which makes 3×E = C
Hint #3
In eq.3, substitute 3×E for C, and 2×E for D: 3×E = A – 2×E Add 2×E to both sides of the equation above: 3×E + 2×E = A – 2×E + 2×E which makes 5×E = A
Hint #4
Substitute 5×E for A in eq.6: F = 5×E + E which makes F = 6×E
Hint #5
Substitute 3×E for C, and 6×E for F in eq.2: B = 3×E + 6×E which makes B = 9×E
Solution
Substitute 5×E for A, 9×E for B, 3×E for C, 2×E for D, and 6×E for F in eq.1: 5×E + 9×E + 3×E + 2×E + E + 6×E = 26 which simplifies to 26×E = 26 Divide both sides of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 5×E = 5 × 1 = 5 B = 9×E = 9 × 1 = 9 C = 3×E = 3 × 1 = 3 D = 2×E = 2 × 1 = 2 F = 6×E = 6 × 1 = 6 and ABCDEF = 593216