Puzzle for August 17, 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.2, replace C with D + E (from eq.3): B + D = D + E + E which becomes B + D = D + 2×E Subtract D from both sides of the above equation: B + D – D = D + 2×E – D which makes B = 2×E
Hint #2
In eq.4, replace B with 2×E: A – 2×E – E = 2×E + E which becomes A – 3×E = 3×E Add 3×E to both sides of the equation above: A – 3×E + 3×E = 3×E + 3×E which makes A = 6×E
Hint #3
In eq.5, substitute 6×E for A, and 2×E for B: D = 6×E + 2×E which makes D = 8×E
Hint #4
Substitute 8×E for D in eq.6: E + F = 8×E Subtract E from each side of the equation above: E + F – E = 8×E – E which makes F = 7×E
Hint #5
Substitute 8×E for D in eq.3: C = 8×E + E which makes C = 9×E
Solution
Substitute 6×E for A, 2×E for B, 9×E for C, 8×E for D, and 7×E for F in eq.1: 6×E + 2×E + 9×E + 8×E + E + 7×E = 33 which simplifies to 33×E = 33 Divide both sides of the above equation by 33: 33×E ÷ 33 = 33 ÷ 33 which means E = 1 making A = 6×E = 6 × 1 = 6 B = 2×E = 2 × 1 = 2 C = 9×E = 9 × 1 = 9 D = 8×E = 8 × 1 = 8 F = 7×E = 7 × 1 = 7 and ABCDEF = 629817